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A178819 Pascal's prism (3-dimensional array) read by folded antidiagonal cross-sections: (h+i; h, i-j, j), h >= 0, i >= 0, 0 <= j <= i.

%I A178819 #22 Jan 17 2020 10:43:20
%S A178819 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,
%T A178819 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,
%U A178819 30,15,20,60,60,20,15,60,90,60,15,6,30,60,60,30,6,1,6,15,20,15,6,1
%N A178819 Pascal's prism (3-dimensional array) read by folded antidiagonal cross-sections: (h+i; h, i-j, j), h >= 0, i >= 0, 0 <= j <= i.
%C A178819 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}.
%H A178819 H. J. Brothers, <a href="https://doi.org/10.1017/S0025557200004447">Pascal's prism</a>, The Mathematical Gazette, 96 (July 2012), 213-220.
%H A178819 H. J. Brothers, <a href="http://www.brotherstechnology.com/math/pascals-prism.html">Pascal's Prism: Supplementary Material</a>
%F A178819 a_(h, i, j) = (h+i-2; h-1, i-j, j-1), h >= 1, i >= 1, 1 <= j <= i.
%F A178819 Recurrence:
%F A178819 For P_h, element a is given by: a_(1, 1) = 1; a_(i, j) = ((i+h-2)/(i-1)) (a_(i-1, j) + a_(i-1, j-1)).
%e A178819 Prism begins (levels 1-4):
%e A178819 1
%e A178819 1 1
%e A178819 1 2 1
%e A178819 1 3 3 1
%e A178819 1
%e A178819 2 2
%e A178819 3 6 3
%e A178819 4 12 12 4
%e A178819 1
%e A178819 3 3
%e A178819 6 12 6
%e A178819 10 30 30 10
%e A178819 1
%e A178819 4 4
%e A178819 10 20 10
%e A178819 20 60 60 20
%t A178819 end = 5; Column/@Table[Multinomial[h, i-j, j], {h, 0, end}, {i, 0, end}, {j, 0, i}]
%Y A178819 Level 1 = A007318.
%Y A178819 Level 2 = A003506.
%Y A178819 Level 3 = A094305.
%Y A178819 Level 4 = A178820.
%Y A178819 Level 5 = A178821.
%Y A178819 Level 6 = A178822.
%Y A178819 Sums of shallow diagonals for each level correspond to rows of square A037027.
%Y A178819 Contains A109649 and A046816.
%Y A178819 P<n, n, n> = A000984.
%Y A178819 P<n, 2n-1, n> = A006480.
%Y A178819 P<n, 3n-2, n> = A000897.
%Y A178819 P<3n-2, 3n-2, n> = A113424.
%K A178819 easy,nonn,tabf
%O A178819 0,6
%A A178819 _Harlan J. Brothers_, Jun 16 2010
%E A178819 Keyword tabf by _Michel Marcus_, Oct 22 2017