A142467 Triangle T(n,m) read by rows: T(n,m) = Product_{i=0..6} binomial(n+i,m)/binomial(m+i,m).

1, 1, 1, 1, 8, 1, 1, 36, 36, 1, 1, 120, 540, 120, 1, 1, 330, 4950, 4950, 330, 1, 1, 792, 32670, 108900, 32670, 792, 1, 1, 1716, 169884, 1557270, 1557270, 169884, 1716, 1, 1, 3432, 736164, 16195608, 44537922, 16195608, 736164, 3432, 1

Offset: 0

Author

Roger L. Bagula, Sep 20 2008

Keywords

• nonn
• easy
• tabl

Comments

Triangle of generalized binomial coefficients (n,k)A342889.%20-%20_N.%20J.%20A.%20Sloane">7; cf. A342889. - _N. J. A. Sloane, Apr 03 2021

Examples

			Triangle begins as:
  1;
  1,     1;
  1,     8,       1;
  1,    36,      36,         1;
  1,   120,     540,       120,         1;
  1,   330,    4950,      4950,       330,         1;
  1,   792,   32670,    108900,     32670,       792,         1;
  1,  1716,  169884,   1557270,   1557270,    169884,      1716,       1;
  1,  3432,  736164,  16195608,  44537922,  16195608,    736164,    3432,    1;
  1,  6435, 2760615, 131589315, 868489479, 868489479, 131589315, 2760615, 6435, 1;
		

Links

• Seiichi Manyama, Rows n = 0..139, flattened
• Johann Cigler, Pascal triangle, Hoggatt matrices, and analogous constructions, arXiv:2103.01652 [math.CO], 2021.
• Johann Cigler, Some observations about Hoggatt triangles, Universität Wien (Austria, 2021).

Crossrefs

Cf. A001263, A005365 (row sums), A056939, A056940, A056941.

Triangles of generalized binomial coefficients (n,k)_m (or generalized Pascal triangles) for m = 1,...,12: A007318 (Pascal), A001263, A056939, A056940, A056941, A142465, A142467, A142468, A174109, A342889, A342890, A342891.

Programs

Magma

[(&*[Binomial(n+j,k)/Binomial(k+j,k): j in [0..6]]): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 13 2022

Mathematica

T[n_, k_]:= Product[Binomial[n+j, k]/Binomial[k+j, k], {j,0,6}];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Nov 13 2022 *)

PARI

T(n, k) = prod(j=0, 6, binomial(n+j, k)/binomial(k+j, k)); \\ Seiichi Manyama, Apr 01 2021

SageMath

def A142467(n,k): return product(binomial(n+j,k)/binomial(k+j,k) for j in (0..6))
flatten([[A142467(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 13 2022

Formula

T(n,m) = A142465(n,m)*binomial(n+6,m)/binomial(m+6,m).

Sum_{k=0..n} T(n, k) = A005365(n).

Extensions

Edited by the Associate Editors of the OEIS, May 17 2009