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Pascal’s Prism: Supplementary Material

(As referenced in “Pascal’s Prism,” The Mathematical Gazette, Vol. 96, July 2012)

Harlan J. Brothers

Brothers Technology, LLC

harlan@brotherstechnology.com

1 Recursive deﬁnition

Using a “level” index hin the recursive relation

a(1,1) = 1; a(i, j)=i+h−2

i−1(a(i−1, j)+a(i−1, j −1)) (1)

one can generate a family of related triangles Thfor levels h={1,2,3, . . . , n}.

Figure 1: The ﬁrst six levels of Pascal’s prism.

Figure 1shows the ﬁrst six rows of each of the ﬁrst six triangles T(1..6), wherein T1is

Pascal’s triangle. These Thcan be arranged sequentially into a 3-dimensional prismatic

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array wherein element a(i, j)of This denoted by a(h, i, j). We refer to the inﬁnite set of

these sequentially arranged triangles as “Pascal’s prism,” denoted by P. Furthermore, in

the manner of a vector-valued function, a sequence of length kthrough Pis deﬁned by

Phh(n), i(n), j(n)ifor n={1, 2, 3, . . . , k}. Thus, for example, with k= 6,

Ph1, n + 1,2i=Ph1, n + 1, ni=Ph2, n, 1i=Ph2, n, ni=Phn, 2,1i={1, 2, 3, 4, 5, 6}.

Higher-ordered paths can also be deﬁned in the same manner. The utility of this vector-

valued notation is demonstrated in Section 3.

2 Explicit deﬁnition

In addition to the recursive approach in (1), Pascal’s prism can be explicitly deﬁned by

the multinomial array h+i

h, i−j, j, h ≥0, i ≥0,0≤j≤i, wherein element a(h,i,j)=

h+i−2

h−1, i−j, j−1. This can be visualized in terms of the ﬁgurate number triangle [10],

F=

10000· · ·

11000· · ·

12100· · ·

13310· · ·

14641· · ·

.

.

..

.

..

.

..

.

..

.

....

and the matrix enumerating the values of the multichoose function n

k, n > 0 [11],

L=

1 1 1 1 1 · · ·

1 2 3 4 5 · · ·

1 3 6 10 15 · · ·

1 4 10 20 35 · · ·

1 5 15 35 70 · · ·

.

.

..

.

..

.

..

.

..

.

....

.

To generate P, we consider Fand Leach as a collection of column vectors. For F, vectors

jk=n

k−1,k≥1, n={0, 1, 2, ...}. For L, “level” vectors jh=n+h−1

h−1,h≥1, n=

{1, 2, 3, ...}.

Next, deﬁne a threaded Hadamard product, denoted by “h◦i”, such that for columns Aj

in m×n matrix A, and columns Bjin m×p matrix B, an m×p×n array is produced:

Ah◦iB={{A1◦B1,A1◦B2,..., A1◦Bp},(2)

{A2◦B1,A2◦B2,..., A2◦Bp},· · · ,{An◦B1,An◦B2,..., An◦Bp}} .

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Then,

Lh◦iF=P.(3)

Unlike the Hadamard product, the threaded Hadamard product is non-commutative.

It is interesting to note that the entire 3-dimensional array Pcan also be described in

terms of the iterated convolution of the simplest sequence of positive numbers with itself. Let

either row or column v0={1,1,1,1,1, ...}and vn= v0⊗v(n−1). Then L={v0, v1, v2,. . . , vn}

and Fis formed from its padded skew diagonals.

3 Some sample sequences

Using the deﬁnition of the multinomial function, it is easy to show that, for n={1,2,3,...},

a(h, n+j−1, j)=a(n, h+j−1, j ). Thus any column jbelonging to an individual level can be ex-

pressed as a pillar that orthogonally traverses levels. While each level can be studied in its

own right, we will only consider a sample of sequences that traverse the diagonals of Pas a

whole.

First, Pappears to oﬀer a framework for uniting many related triangular and square

arrays. For instance, sequences of the form Phn, n +k, ni, k ≥1, relate to the enumeration

of Schr¨oder paths [12] and constitute the columns of OEIS sequence A104684 and its mirror

image, A063007, wherein column jis given by (2n+j−1)!

(j−1)!n!2, for n={1,2,3, . . . }.

Sequences of the form Phn, n +k, n +ki, k ≥1, relate to the expansion of Chebyshev

polynomials [8] and the enumeration of Dyck paths [9]. They constitute the non-zero entries

in the columns of A100257 (see also A008311).

Sequences of the form Phk(n−1) + 1, n, ni, k ≥1, constitute the rows of A060539,

the triangle enumerating nk

k. Its main diagonal (or central values) A014062 are given by

Phn2−n+ 1, n + 1, n + 1i.

Palso contains many speciﬁc sequences of interest. For example, the following sequences

appear in Ramanujan’s theory of elliptic functions [1]:

•Phn, 2n−1, ni={1,6,90,1680,34650, . . . }, associated with signature 3 [5]

•Phn, 3n−2, ni={1,12,420,18480,900900, . . . }, associated with signature 4 [2]

•Ph3n−2,3n−2, ni={1,60,13860,4084080,1338557220, . . . }, associated with signa-

ture 6 [6].

The sequence associated with signature 2 is simply Phn, n, ni2={1,4,36,400,4900, . . . }[3].

Because it is itself composed of a family of triangles, the series of sequences for 1) the

row sums, and 2) the row products of the respective levels of Pcan be compiled in to master

rectangular arrays (see Figure 2).

3

1 2 4 8 16 · · ·

1 4 12 32 80 · · ·

1 6 24 80 240 · · ·

1 8 40 160 560 · · ·

1 10 60 280 1120 · · ·

.

.

..

.

..

.

..

.

..

.

....

1 1 2 9 96 · · ·

1 4 54 2304 300000 · · ·

1 9 432 900000 72900000 · · ·

1 16 2000 1440000 5042100000 · · ·

1 25 6760 13505625 161347200000 · · ·

.

.

..

.

..

.

..

.

..

.

....

Figure 2: Array of row sums (left) and row products (right) for the ﬁrst 5 levels of P.

While only the ﬁrst two rows and columns of the row products array are familiar se-

quences, in the case of the row sums array, we ﬁnd a rich collection of well-known sequences.

The ﬁrst ﬁve rows correspond respectively to A000079,A001787,A001788,A001789, and

A003472, and for row hare given by ah(n)= 2(n−h)n

h, n ≥h. The ﬁrst ﬁve columns cor-

respond respectively to sequences A000012,A005843,A046092,A130809, and A130810 and

for column jare given by aj(n)= 2jn

n−j, n ≥j.

In addition, its main diagonal is given by A059304, the ﬁrst superdiagonal by A069723

(beginning with the second term), and the ﬁrst subdiagonal by A069720. The skew diagonals

together form A013609, the triangle which enumerates the coeﬃcients in the expansion of

(1 + 2x)n.

Finally, in examining the overall structure of P, we ﬁnd the sequence of sums of the

shallow diagonals of each level correspond to consecutive convolutions of the Fibonacci series

with itself. For level h, the sums are given by the generating function 1/(1 −x−x2)hand

collectively form the rows of the skew Fibonacci-Pascal triangle [7].

References

[1] B. C. Berndt, S. Bhargava and F. G. Garvan, Ramanujan’s theories of elliptic functions

to alternative bases , Transactions of the American Mathematical Society,347, (1995),

4163–4244.

[2] N. J. A. Sloane, “Sequence A000897.”

http://oeis.org/A000897.

[3] N. J. A. Sloane and M. Somos, “Sequence A000984.”

http://oeis.org/A000984.

[4] N. J. A. Sloane, “Sequence A003506.”

http://oeis.org/A003506.

[5] N. J. A. Sloane, “Sequence A006480.”

http://oeis.org/A006480.

[6] M. Somos, “Sequence A113424.”

http://oeis.org/A113424.

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[7] F. van Lamoen, “Sequence A037027.”

http://oeis.org/A037027.

[8] E. W. Weisstein, “Chebyshev Polynomial of the First Kind.” From MathWorld - A

Wolfram Web Resource.

http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html.

[9] E. W. Weisstein, “Dyck Path.” From MathWorld - A Wolfram Web Resource.

http://mathworld.wolfram.com/DyckPath.html.

[10] E. W. Weisstein, “Figurate Number Triangle.” From MathWorld - A Wolfram Web

Resource.

http://mathworld.wolfram.com/FigurateNumberTriangle.html .

[11] E. W. Weisstein, “Multichoose.” From MathWorld - A Wolfram Web Resource.

http://mathworld.wolfram.com/Multichoose.html.

[12] E. W. Weisstein, “Schr¨oder Number.” From MathWorld - A Wolfram Web Resource.

http://mathworld.wolfram.com/SchroederNumber.html.

2000 Mathematics Subject Classiﬁcations: Primary 05A10; Secondary 28A80.

Keywords and phrases: binomial, coeﬃcient, combinatorics, e, fractal, logarithm, Pascal,

Sierpinski, tetrahedron.

Concerned with OEIS sequences: A000012,A000079,A000897,A000984,A001787,A001788,

A001789,A003472,A003506,A005843,A006480,A008311,A013609,A014062,A037027,

A046092,A059304,A060539,A063007,A069720,A069723,A100257,A104684,A113424,

A130809, and A130810.

c

2010 Harlan J. Brothers

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