A178822 - OEIS

A178822

Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.

1, 6, 6, 21, 42, 21, 56, 168, 168, 56, 126, 504, 756, 504, 126, 252, 1260, 2520, 2520, 1260, 252, 462, 2772, 6930, 9240, 6930, 2772, 462, 792, 5544, 16632, 27720, 27720, 16632, 5544, 792, 1287, 10296, 36036, 72072, 90090, 72072, 36036, 10296, 1287

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OFFSET

0,2

COMMENTS

The product of A000389 and Pascal's triangle (A007318). Level 6 of Pascal's prism (A178819) read by rows: (i+5; 5, i-j, j), i >= 0, 0 <= j <= i.

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

H. J. Brothers, Pascal's prism, The Mathematical Gazette, 96 (July 2012), 213-220.

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

FORMULA

T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.

For element a_(h, i, j) in A178819: a_(6, i, j) = (i+4; 5, i-j, j-1), i >= 1, 1 <= j <= i.

G.f.: 1/(1 - x - x*y)^6. - Ilya Gutkovskiy, Mar 20 2020

EXAMPLE

Triangle begins:

6, 6;

21, 42, 21;

56, 168, 168, 56;

126, 504, 756, 504, 126;

MATHEMATICA

Table[Multinomial[5, i-j, j], {i, 0, 9}, {j, 0, i}]//Column

Table[Binomial[n + 5, 5]*Binomial[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Nov 25 2017 *)

PROG

(Magma) /* As triangle */ [[Binomial(n+5, 5)*Binomial(n, k): k in [0..n]]: n in [0..10]]; // Vincenzo Librandi, Oct 23 2017

(PARI) for(n=0, 10, for(k=0, n, print1(binomial(n+5, 5)*binomial(n, k), ", "))) \\ G. C. Greubel, Nov 25 2017

CROSSREFS

Cf. A000389, A007318, A178819, A178820, A178821.

Rows sum to A054849, shallow diagonals sum to A001874.

Sequence in context: A188273 A185786 A341245 * A253069 A255460 A255464

Adjacent sequences: A178819 A178820 A178821 * A178823 A178824 A178825

KEYWORD

easy,nonn,tabl

AUTHOR

Harlan J. Brothers, Jun 19 2010

STATUS

approved