Spreadsheets, Power Series, Generating Functions, and Integers

Spreadsheets, Power Series, Generating Functions, and IntegersAuthor(s): Donald R. SnowSource: The College Mathematics Journal, Vol. 20, No. 2 (Mar., 1989), pp. 143-152Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2686271 .

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COMPUTER

CORNER

Edited by John Blattner

In this column, readers are encouraged to share their expertise and experiences with computers as they relate to college mathematics. Articles may illustrate how computers can be used to enhance pedagogy, solve problems, or model real-life situations. Readers are also invited to submit interesting (not necessarily original) algorithms in a structured language or pseudocode, with explanatory text to make the purpose and validity clear.

All manuscripts for this column should be prepared according to the guidelines on the inside front cover and sent to:

Eugene Herman Department of Mathematics Grinnell College Grinnell, IA 50112

Spreadsheets, Power Series, Generating

Functions, and Integers

Donald R. Snow

Donald R. Snow was born and raised in Los Angeles, and received two bachelor's degrees from the University of Utah, and two master's degrees and a Ph.D. (math, 1965) from Stanford. He has been on the faculty of Brigham Young University since 1969, with a sabbatical to the University of Waterloo and a Fulbright appointment in Lima, Peru. His research interests are in functional equations, combina? torics, calculus of variations, history of math, and computers in math. He taught a minicourse on using computer spread? sheets in math at the Summer 1987 MAA meeting.

The generating function for the Fibonacci numbers, 1, 1, 2, 3, 5, 8, 13,... is the rational function G(x) = 1/(1 - x - x2) = 1 + lx 4- 2x2 4- 3x3 + 5x4 4- ... . The co? efficients of both G(x) and its reciprocal appear in the following table, which was

generated on a spreadsheet.

143



Power

G(xy3 G(xy2 G(x)-1 G(x)?

G(x)1

G(x)2 G(xf

-3 -2 -1

2 3 X XJ

0 5 -1 2 -1 0

10 22

0-3-1 0 10 0 0 0 0 0 0

3 5

20 51

38 111

13

71 233

21

130 474

0 0 0

34

235 942

0 0 0

55

420 1836

0 0 0

89

744 3522

0 0 0

144

1308 6666

0 0 0

233

2285 12473

The numbers in the box were the only numbers actually entered; the spreadsheet generated all the rest. The Fibonacci numbers occur on the row labeled G(x)1 and the coefficients of the reciprocal of the generating function are produced on row

G(x)-1. The two starting entries can be changed to give other Fibonacci-type sequences on row G(x)1. For example, if the first two Lucas numbers, 1 and 3, go in the box, then the corresponding Lucas number sequence would appear on row

G(x)1, with the reciprocal of its generating function on row G(x)-1. How does the

spreadsheet do all this? Two previous articles in this journal provide background for this one. Watkins

[10] discusses the theory and uses of generating functions with many examples from combinatorics and number theory. Arganbright [2] discusses the use of spreadsheet programs in mathematics. (Both of these articles use the Fibonacci sequence as an

example, but from different perspectives.) We will assume familiarity with at least the basic ideas about generating functions and the use of spreadsheets as discussed in these articles. Further information on generating functions occurs in most books on combinatorics [3], [4], [5], [8]. There are many spreadsheet programs available for

microcomputers, some very inexpensive yet powerful (e.g., VP-Planner/Academic [9], which is the spreadsheet used to generate the tables in this article).

The Table Algorithm

Suppose G(x) = a0 + axx + a2x2 + ??? =Y.f=0aixi is a formal power series. We construct a table of the coefficients C(n, i) of the powers, positive and negative, of

G(x) as follows:

G(xy] G(x)? G(x)1 G(x)2 G(x)3

G(x)"

C(-2,0) C(-1,0) C(0,0) C(l,0) = a0 C(2,0) C(3,0)

C(?,0)

C(-2,l) C(-l,l) C(0,1) C(l,l) = *i C(2,l) C(3,l)

C(?,l)

C(-2,2) C(-l,2) C(0,2) C(l,2) = a2 C(2,2) C(3,2)

C(n,2)

C(-2,3) C(-l,3) C(0,3) C(l,3) = ?3 C(2,3) C(3,3)

C(/f,3)

The nth row in the table consists of the coefficients of the nth power of the formal

power series G(x). To determine a simple algorithm that generates this table, we

144



consider the (m + n)th row. Since G{x)m + n = G(x)mG(x)n, we have

00

E C(n,k)xk ? C(m + n, i)xl =

/ = 0 ? C(m,j)xj

7 = 0 k = 0

I i = 0

E C(m,j)C(n,i-j) 7 = 0

x .

So the table elements satisfy

C(/w+ /?,/) = E C(m,j)C(n,i-j). 7 = 0

(1)

(2)

Thus, the convolution ("backwards" inner product) of the mth and nth rows yields the (m + n)th row. Since the first row is the original power series, we set m = 1 and use C(l, j) =

aj to obtain

C(n + l,i)= ? ajC(n,i-j). 7 = 0

(3)

This linear recurrence relation, which gives the (n 4- l)st row of the table in terms of the first and nth rows, is a generalization of the Pascal property that generates Pascal's triangle. It reduces to the Pascal property in the special case G(x) = l + x.

The linear recursion (3) can be visualized by considering the template

atC(n -

1,0) + ? ? ? +a2C(n -l,i- 2) + axC{n - 1, i - 1) + a0C(n

- 1, /)

= C(n,i). (3a)

Using (3)-(3a) and working downward and to the right, we generate rows 2,3,... of

our table. For this algorithm to produce the sequence a0, av... on row 1, row 0

must be 1,0,0,.... So we enter 1,0,0,... on row 0. Various shortcuts are possible in

generating parts of the table. For example, if only the even order rows are desired, we first find row 2 from row 1. Then setting m = 2 in (2) gives

C(? + 2,i)= ZC{2,j)C{n,i-j), 7 = 0

so the coefficients in the two-row recurrence relation are the elements from row 2.

Similarly, every third row can be obtained via the coefficients C(3, j) and so on.

Algorithm (3)-(3a) also gives the product of the given power series Lf=0atxl with

any other power series LJIq^x', since the coefficient of xl in the product series

2*i = ftCjX IS

flA + fli-A+ ??' +a2bi_2 + albi_l + a0bi

Since the algorithm gives the product of any power series with the given one, the rows for negative n are obtained by using the algorithm backwards. That is, since the original algorithm is a relation involving C(n + 1, i) in terms of the elements on

145



the row above and to the left of it, C(nJ) can be found in terms of the element

C(n 4- 1, i) below it and all those elements to the left of it on its same row. This recurrence relation is

C(/i, i) = (l/a0)[C(n + 1, /) -

aiC(n, / - 1) -

a2C(n, i-2)-atC(n90)]

(4)

or, in template form

(l/*o) -atC(n90)- ??? -fl2C(/7,/-2)-^1C(/7,/-l) \=C(/i,z)

+ C(/i + l,/)

Thus, rows -1, - 2, - 3,... can be generated by working upward and to the right. The completed table then satisfies template (3a) everywhere.

After the spreadsheet makes the computations (the hard part) for us, we look for

patterns in the table and from these obtain the generating function. Various types of

patterns, including both rational and nonrational generating functions, are illus? trated later. Following the examples we discuss some additional uses of the tables.

Implementation of the Algorithm on a Spreadsheet

Most spreadsheets have their columns labeled A, B, C,..., and their rows num? bered 1,2,3,..., so the cell in column D and row i is designated Di. (This labelling is the reverse of matrix notation, since the column label precedes the row number.) Spreadsheet programs allow arithmetic operations involving any cell or combination of cells. The table algorithm is implemented with a spreadsheet in the following way. (The coefficients of G(x)1 are entered on row 11 here, but any other row could have been used.)

Column Row A B C D E F

9 A9 B9 C9 D9 E9 10 1 0 0 0 0 11 a0 ax a2 a3 a4 12 A12 B12 C12 D12 13 A13 B13 C13 D13

The sequence 1,0,0,... is entered once and for all on row 10. In the examples given, only the first 13 terms (ending with the coefficients of x12) are displayed, but the number of terms could be made as large as desired. Likewise, only three negative power rows and three positive power rows are shown.

Examples of two of the cell formulas, one below and one above the main row, are

Cell D13: +A$11*D12 + B$11*C12 + C$11 *B12 + D$11*A12

Cell E9: + (E10 - B$ll * D9 - C$11 * C9 - D$ll * B9 - E$ll * A9)/A$ll.

146



The notation here means the value to be computed for cell D13 is the sum of the

products of the values of cells All, Bll, Cll, and Dll with those in D12, C12, B12, and A12, respectively. This is the convolution of the first parts of rows 11 and 12. For the rows below row 11 the corresponding column formula only needs to be entered once in the cell on row 12 and then copied to the cells below by the

spreadsheet, which makes the appropriate changes for the row labels. Similarly, the formulas above can be copied upwards. After all of the formulas have been entered, the spreadsheet is saved on the disk so it can be recalled at any time. To use it, we enter the desired sequence on row 11; all the entries except those in rows 10 and 11 are then computed. The resulting spreadsheet can be printed on paper or to disk for use in another computation or word processing program.

Spreadsheets allow numbers in integer format up to a certain limit (e.g., 15 digits in VP-Planner), so the computations are exact up to that precision. Hence, if the entered power series has only integer coefficients, the lower part of the table will contain only integers. Since the upper part of the computation (negative powers of

G(x)) involves a division by a0, fractions may arise if a0 =? ?1. They may appear everywhere, if the power series has fractional coefficients. It would be nice to have a

spreadsheet with fractional form available so the output would be exact throughout the spreadsheet. Fractions could be handled by using two rows, one for the numerators and one for the denominators, but this would complicate the formulas

considerably, since each fraction would need to be reduced to simplest terms at each

step. In the following examples the sequence numbers in parentheses are from Sloane

[6], which also contains additional references.

Fibonacci numbers (Sequence #256): 1,1,2,3,5,8,... These are obtained from the recursion Fn + 2 = Fn+l + Fn, F0 = F1 = l (see the first table). The Fibonacci numbers can be entered individually on row 11 or the first two cells can be loaded with 1 -?All and 1 -^ Bll and the formula Cll = All + Bll entered and copied to the rest of the row. When the spreadsheet is run, row 9 contains the coefficients of the

polynomial 1-x-x2, so the generating function is G(x) = 1/(1 - x - x2), the well-known Fibonacci number generating function. If the coefficients of the

polynomial 1 - x - x2 are entered on row 11, the Fibonacci numbers appear on row 9, since they are the coefficients in the expansion of 1/(1 - x - x2). Thus the table

algorithm produces the generating function from the sequence and the sequence from the generating function. If the Fibonacci numbers are entered starting with an extra 1 (that is, 1,1,1,2,3,5,8,...), another pattern arises in the table, which gives the generating function for the sequence starting with F0 = F1 = F2 = l. Using 1,3,4,7,... we obtain the generating function for the Lucas numbers, which satisfy the same recursion relation but have initial conditions L0 = 1, Lx = 3.

For the rest of the examples, we will refer to the rows in the spreadsheet by the

power of G(x) involved.

Catalan numbers (Sequence #577): 1,2,5,14,42,132,429,1430,... Catalan numbers arise in several places in combinatorics (e.g. the number of ways to parenthesize an

expression) and are given by (2ww)/(? + 1). When they are entered on row 1 starting

147



with an additional 1 the table algorithm gives:

Power

G(x)-> G(xy2 G(xyl G(x)? G(x)1 G(x)2 G(x)3

-1 -2 -2

5 14 28

-3 -5 -5

14 42 90

-9 -14 -14

42 132

-28 -42 -42

132 429

-90 -132 -132

-297 -1001 -429 -1430 -429 -1430

x10 x11 x12

-3432 -11934 -41990 -4862 -16796 -58786 -4862 -16796 -58786

429 1430 1430 4862

4862 16796

16796 58786

297 1001 3432 11934 41990 149226

58786 208012 208012 742900 534888 1931540

We note several patterns here, and we can get the generating function in several

ways. For example, shifting row 2 one cell to the right and appending 1 on the left

gives row 1. So

xG(x)2 4-1 = G(x) or xG(x)2 - G(x) 4-1=0,

a quadratic equation in G(x) with roots (1 + ]/l - Ax )/2x. The negative sign must be used so that G(x) is defined at x = 0. Therefore, G(x) = (1 ? y/1 ? 4x )/2x. Note how rows n and ?n are related for n = 1, 2, and 3. A reasonable conjecture is that this pattern holds for all n.

If we shift the Catalan numbers to the left on row 1 and leave out the first 1, a different pattern emerges. There are still relationships between rows n and - n but

they are different now.

Power

G{x)~3 G(xy2 G(xyl G(x)? G(x)1 G(x)2 G(x)3

-6 -4 -2

2 4

9 2

-1

5 14

-2 0 0 -1

-2 -5

14 42 48 165

x5 x6

0 -1 -4 -14

-14 -42

132 429 572 2002

-6 -48

-132

-27 -165

-110 -572

-429 -1430

x10

-429 -2002 -4862

x11 x12

-1638 -6188 -7072 -25194

-16796 -58786

1430 4862 7072 25194

16796 90440

58786 208012 742900 326876 1188640 4345965

6 27 110 429 1638 6188 23256 87210 326876 1225785 4601610 *******

(The asterisks in the table mean that not enough spaces were allowed to print all the

digits of the integer.) Since the shifted Catalan number sequence on row 1 here occurs on row 2 of the original table, the generating function of the shifted sequence is the square of the generating function of the original sequence. As in the case of the Fibonacci numbers, the original sequence can be started in various places on row 1 to recognize a pattern.

The natural numbers (Sequence # 173): 1,2,3,...

Power

G{x)~3 G(xy2 G(xyl G(xf G(x? G(x)2 G(x)3

15 6 1

2 3 4 10 6 21

-20 15 -4 1

0 0

4 5 20 35 56 126

-6 1 0 0 0 0 0 0 0

56 252

7 84

462 120 792

0 0 0

9 165

1287

0 0 0

10 220

2002

0 0 0

11 286

3003

0 0 0

12 364

4368

0 0 0

13 455

6188

From the pattern on row ?1, we see that the generating function is G(x) =

148



1/(1 - 2x + x2) = 1/(1 - x)2. This expansion is usually obtained in calculus texts

by twice differentiating the power series for -ln(l - x).

The "Tribonacci" numbers (Sequence #296): 1,2,3,6,11,20,37,... These are ob? tained from Fn+3 = Fn+2 + Fn+l + Fn, with F0 = 1, Fx = 2, F2 = 3.

Power

G(x)-2 G(x)-1 G(x)? G(x)1 G(x)2 G(x)3

-4 -2

2 4

15 6 1

3 10 21

x

-26 45 13 2

6 11 24 55 62 168

-72 -16 -2

20 120

103 20 2

37 256

-144 -24 -2

192 28 2

-248 -32 -2

230 2252

68 125 536 1105

2418 5517 12324 27057 58548

312 36 2

423 4546

x11

-384 -40 -2

778 9104

426 1032

The pattern in row -1 is related to the geometric series l-x + x2-x3+ ??

1/(1 + x), and we have

1/G(x) = 1 - 2x + x2 - 2x3(l - x + x2 - x3 + ? ? ? )

= (l-x)2-2x3/(l+x).

Thus, G(x) = (1 + x)/[(l + x)(l - x2) - 2x3].

Convolved Fibonacci numbers (Sequence # 1595): 1,5,19,65,210,654,1985,...

Power .x0

G(xy3 G(xy2 G(xyl G(x)? G(x)1 G(x)2 G(x)3

x12

464 44 2

1431 18111

125144

-15 93 -305 561 -10 37 -60 38 -5 6 0 1

5 19 10 63 15 132

65 210

x

-567 -8

1

654

300 4 1

-96 4 1

15 5 1

1985 5911 50305 144516

15 8 1

411900 1166209

22 9 1

17345 320 1431 5878 22715 83824 298350 1031408 3481038 11514400 37439389 890 5088 25962 121849 536259 2242425 8993891 ******* ******* ******

We have another geometric series in row ?1. Thus,

1/G(x) = 1 - 5x + 6x2 + x4(l + x + x2 + ? ? ? )

= (1 -

2x)(l -

3x) + x4/(l "

*)?

So

G(x) = (1 -

x)/[(l - x)(l

- 2x)(l

- 3x) + x4].

Central binomial coefficients (Sequence #1144): 1, 3, 10, 35, 126, 462,

I7i6,...(*-;i),...

-3 Power

G(x) G(x)-2 G(xyl G(x)? G(x? G(x)2 G(x)3

-9 24 -6 7 -3 -1

-15 2

-2

-3 6

-14

-4 14

-42 36

-132

-9 99

-429

-11 286

-1430

x10

0 858

-4862

78 2652

-16796

xu

442 8398

-58786

3 10 35 126 462 1716 6435 24310 92378 352716 1352078 5200300 6 29 130 562 2380 9949 41226 169766 695860 2842226 11576916 47050564 9 57 312 1578 7599 35401 161052 719790 3173090 13836426 59803104 *******

149



Since the Catalan numbers, C?, appear in row -1, we can obtain a formula for them in terms of the central binomial coefficients. First express G(x)~l in terms of the generating function F(x) = (1 - /l - 4x )/2x for the Catalan numbers as follows:

G(x)_1 = l-3x-x(x + 2x2+5x34- -?-) = l-3x-x(F(x)-l)

= (1 - 4x + y[T::rAx~)/2.

By inverting both sides of the above equation and rationalizing the denominator, we

get an expression for G(x) in terms of F(x). Specifically,

2 G(x) =

1 - 4x + \/l-4x

-1 4- 4x + \/l-4x ~

2x(l-4x)

2-F(x)

l-4x

Now we obtain the standard formula for the nth Catalan number by comparing the coefficients on xn in the equation 2 - F(x) = (1 ? 4x)G(x) to get

After we have found the generating function, the table algorithm can be used to obtain more terms of the sequence. This technique was illustrated earlier in the Fibonacci example. That we have really found the generating function for the original sequence needs to be verified analytically or combinatorially using the definition of the sequence, not just the first few terms, since infinitely many sequences start the same way.

Integers

This algorithm multiplies and divides integers in any number base, since the variable x can be taken to be any positive integer greater than 1 (such as 10 or 2). If the digits of an integer are used as the entries on row 1, the positive power rows give the positive powers of the integer and the negative power rows give the powers of its

reciprocal. The only change that must be made is the addition of a "carry" algorithm so that, when the cell entry becomes larger than the number base, additional digits will be carried to the next higher power of x. On a spreadsheet the mod function gives the correct entry at a given cell, but a more complicated formula is needed to determine the carry to the entry to the right. The entries in column /

require summing i 4- 1 terms, each of which is a product, so the carry can be quite large. One way to take care of this is to run the algorithm without carrying; then have the spreadsheet calculate the integer C(n, 0) 4- C(n, l)x 4- C(n, 2)x2 + ? ? ? and use its digits as the entries in the ?th row. An algorithm using the mod function can be developed to work to the right on a second pass over the entire spreadsheet or on a copy of it below the original one.

150



Several interesting number theory questions arise here. What interesting results can be obtained if the digits of an integer are not restricted to be less than the number base? We will refer to these as superdigits. Their use eliminates the

uniqueness of representation of an integer in a given number base but allows powers to be computed easily by the table algorithm. It is interesting to study integers whose superdigits are the first n Fibonacci numbers, for example. What do the

negative power rows represent for these integers?

Other Applications of the Table Algorithm

Laurent Expansions. The table algorithm gives the Laurent expansion of a rational function if the substitution z = 1/x is made first. Care must be taken as to which order to enter the coefficients and the powers that are involved. This order becomes clear once the substitution has been made.

Sequences of Functions. The table algorithm works just as well with sequences of functions, e.g. orthogonal polynomials, as with number sequences. This requires using algebra instead of just arithmetic. For example, entering the sequence l,2t, 3t2,4t3,5t4,... on row 1 yields 1,-2t, t2,0,0,... on row -1, so G(t, x) =

1/(1 - 2tx 4- t2x2) is the generating function. Conversely, entering 1, - 2t, t2,0,0,... on row 1 yields the sequence on row ? 1. It is possible to compute these by hand, but a spreadsheet that does symbolic algebra would be very helpful.

Sequences of Matrices. The table algorithm also works for sequences of matrices

A0 + Axx + A2x2 + ? ? ? where A0 is nonsingular. Spreadsheets can be used for this

also, but now a square block of cells is required for each matrix. A spreadsheet that handled matrices as elements would be helpful here.

Miscellaneous Uses. Additional uses of the table algorithm occur wherever powers of a polynomial or power series arise. By using p(x)q(x) = (l/2)[(p + q)2 ? p2 ?

q2], products of any two power series or polynomials can be expressed in terms of the squaring operation only, so a general product can be obtained using the first two rows. Modifications of the table algorithm itself can be made to give products and

quotients of any two power series, and hence, expansions of rational functions.

Also, modifications of it have been made to yield exponential and other types of

generating functions [7].

Conclusions

The table algorithm gives the integral powers, positive and negative, of a polynomial or power series. It has been implemented on a spreadsheet program and used to obtain generating functions for several number sequences such as the Fibonacci and Catalan numbers. These illustrate the discovery of both rational and nonrational

generating functions. The table algorithm can also be used to find sequences from their generating functions. Some of the generating functions we have obtained are new (e.g., Sequence #1595), or at least we can find no reference to them in the literature. The generating functions for about 50 integer sequences selected almost at random from [6] have been obtained this way and, undoubtedly, many are new. We also considered the application of the table algorithm to integers represented in various number bases, since these can be considered as polynomials in the number base.

151



REFERENCES

1. Deane E. Arganbright, Mathematical Application of Electronic Spreadsheets, McGraw-Hill, New York, 1985.

2. _, The electronic spreadsheet and mathematical algorithms, The College Mathematics Journal 15 (1984) 148-157.

3. Richard A. Brualdi, Introductory Combinatorics, North-Holland, New York, 1977. 4. Daniel I. A. Cohen, Basic Techniques of Combinatorial Theory, John Wiley and Sons, New York,

1978. 5. John Riordan, An Introduction to Combinatorial Analysis, John Wiley and Sons, New York, 1958.

Reprinted by Princeton University Press, Princeton, NJ, 1980. 6. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, New York, 1973. 7. Donald R. Snow, Algorithms for Old and New Sequence Generating Functions, Annual Meeting,

American Mathematical Society, Atlanta, Georgia, 6-9 Jan 1988. Abstract in Abstracts of the American Mathematical Society, 9 (Jan 1988) 18.

8. Alan Tucker, Applied Combinatorics, 2nd edition, John Wiley and Sons, New York, 1984. 9. VP-Planner/Academic, Holden-Day, Oakland, CA, 1987, $19.95.

10. William Watkins, Generating functions, The College Mathematics Journal 18 (1987) 195-211.

Y\kk'k\ = (n\) k = \

? + l

1 11

2 2

3 3 3 3

n- 2 n-2 n-2 '

n-l n-l n - I '

n n n

n-2 \n - 2

n-l n-l

n-2

n-l

n n n

? + l-

Submitted by Edward T. H. Wang, Wilfrid Laurier University, Waterloo, Ontario

152



Documents

Spreadsheets, Power Series, Generating Functions, and Integers