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

Offset: 0

Author

Harlan J. Brothers, Jun 19 2010

Keywords

• easy
• nonn
• tabl

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.

Examples

			Triangle begins:
    1;
    6,   6;
   21,  42,  21;
   56, 168, 168,  56;
  126, 504, 756, 504, 126;
		

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

Crossrefs

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

Rows sum to A054849, shallow diagonals sum to A001874.

Programs

Magma

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

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 *)

PARI

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

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