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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. 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 (list; graph; refs; listen; history; text; internal format)
 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 for n = {1, 2, 3, ..., k}. LINKS 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 = A000984. P = A006480. P = A000897. P<3n-2, 3n-2, 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

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Last modified February 19 19:04 EST 2020. Contains 332047 sequences. (Running on oeis4.)