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.

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

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), 213-220.

H. J. Brothers, Pascal's Prism: Supplementary Material

Formula

a_(h, i, j) = (h+i-2; h-1, i-j, j-1), h >= 1, i >= 1, 1 <= j <= i.

Recurrence:

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)).

Example

Prism begins (levels 1-4):
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, i-j, 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 correspond to rows of square A037027.

Contains A109649 and A046816.

P<n, n, n> = A000984.

P<n, 2n-1, n> = A006480.

P<n, 3n-2, n> = A000897.

P<3n-2, 3n-2, n> = A113424.

Sequence in context: A089955 A352942 A180312 * A369174 A355855 A046816

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