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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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