A142465 Triangle T(n,m) read by rows: T(n,m) = Product_{i=0..5} binomial(n+i,m)/binomial(m+i,m).

1, 1, 1, 1, 7, 1, 1, 28, 28, 1, 1, 84, 336, 84, 1, 1, 210, 2520, 2520, 210, 1, 1, 462, 13860, 41580, 13860, 462, 1, 1, 924, 60984, 457380, 457380, 60984, 924, 1, 1, 1716, 226512, 3737448, 9343620, 3737448, 226512, 1716, 1, 1, 3003, 736164, 24293412, 133613766, 133613766, 24293412, 736164, 3003, 1

Offset: 0

Author

Roger L. Bagula, Sep 20 2008, Jan 28 2009

Keywords

• nonn
• easy
• tabl

Comments

Triangle of generalized binomial coefficients (n,k)A342889.%20-%20_N.%20J.%20A.%20Sloane">6; cf. A342889. - _N. J. A. Sloane, Apr 03 2021

The matrix inverse starts

1;

-1, 1;

6, -7, 1

-141, 168, -28, 1;

9911, -11844, 2016, -84, 1;

-1740901, 2081310, -355320, 15120, -210, 1. - R. J. Mathar, Mar 22 2013

Examples

			Triangle begins as:
  1;
  1,    1;
  1,    7,      1;
  1,   28,     28,        1;
  1,   84,    336,       84,         1;
  1,  210,   2520,     2520,       210,         1;
  1,  462,  13860,    41580,     13860,       462,        1;
  1,  924,  60984,   457380,    457380,     60984,      924,      1;
  1, 1716, 226512,  3737448,   9343620,   3737448,   226512,   1716,    1;
  1, 3003, 736164, 24293412, 133613766, 133613766, 24293412, 736164, 3003, 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, A005364 (row sums), A056941, A056940, A056939, A142467.

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

A142465:= func< n,k | (&*[Binomial(n+j,k)/Binomial(k+j,k): j in [0..5]]) >;
[A142465(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 13 2022

Maple

A142465 := proc(n,m)
    mul(binomial(n+i,m)/binomial(m+i,m),i=0..5) ;
end proc; # R. J. Mathar, Mar 22 2013

Mathematica

T[n_, k_]:= Product[Binomial[n+j, k]/Binomial[k+j, k], {j,0,5}];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten

PARI

T(n, k) = prod(j=0, 5, binomial(n+j, k)/binomial(k+j, k)); \\ Seiichi Manyama, Apr 01 2021

SageMath

def A142465(n,k): return product(binomial(n+j,k)/binomial(k+j,k) for j in (0..5))
flatten([[A142465(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 13 2022

Formula

T(n,m) = A056941(n,m)*binomial(n+5,m)/binomial(m+5,m).

Sum_{k=0..n} T(n, k) = A005364(n).

Extensions

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