

A178819


Pascal's prism (3dimensional array) read by folded antidiagonal crossections: (h+i; h, ij, j), h >= 0, i >= 0, 0 <= j <= i.


3



1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 3, 1, 3, 6, 3, 3, 3, 1, 1, 4, 4, 6, 12, 6, 4, 12, 12, 4, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 5, 20, 30, 20, 5, 10, 30, 30, 10, 10, 20, 10, 5, 5, 1, 1, 6, 6, 15, 30, 15, 20, 60, 60, 20, 15, 60, 90, 60, 15, 6, 30, 60, 60, 30, 6, 1, 6, 15, 20, 15, 6, 1
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OFFSET

0,6


COMMENTS

P_h = level h of Pascal's prism where P_1 = Pascal's triangle (A007318) and P_2 = denominators of Leibniz harmonic triangle (A003506). A sequence of length k through P is defined by P<h(n), i(n), j(n)> for n = {1, 2, 3, ..., k}.


LINKS

Table of n, a(n) for n=0..83.
H. J. Brothers, Pascal's prism, The Mathematical Gazette, 96 (July 2012), 213220.
H. J. Brothers, Pascal's Prism: Supplementary Material


FORMULA

a_(h, i, j) = (h+i2; h1, ij, j1), h >= 1, i >= 1, 1 <= j <= i.
Recurrence:
For P_h, element a is given by: a_(1, 1) = 1; a_(i, j) = ((i+h2)/(i1)) (a_(i1, j) + a_(i1, j1))


EXAMPLE

Prism begins (levels 14):
1
1 1
1 2 1
1 3 3 1
1
2 2
3 6 3
4 12 12 4
1
3 3
6 12 6
10 30 30 10
1
4 4
10 20 10
20 60 60 20


MATHEMATICA

end = 5; Column/@Table[Multinomial[h, ij, j], {h, 0, end}, {i, 0, end}, {j, 0, i}]


CROSSREFS

Level 1 = A007318
Level 2 = A003506
Level 3 = A094305
Level 4 = A178820
Level 5 = A178821
Level 6 = A178822
Sums of shallow diagonals for each level corrrespond to rows of square A037027
Contains A109649 and A046816
P<n, n, n> = A000984
P<n, 2n1, n> = A006480
P<n, 3n2, n> = A000897
P<3n2, 3n2, n> = A113424
Sequence in context: A087775 A089955 A180312 * A046816 A301475 A138328
Adjacent sequences: A178816 A178817 A178818 * A178820 A178821 A178822


KEYWORD

easy,nonn,tabf


AUTHOR

Harlan J. Brothers, Jun 16 2010


EXTENSIONS

Keyword tabf by Michel Marcus, Oct 22 2017


STATUS

approved



