A174109 Triangle read by rows: T(n, k) = c(n, q)/(c(k, q)*c(n-k, q)), where c(n, q) = Product_{j=1..n} (j+q)!/(j-1)! and q = 8.

1, 1, 1, 1, 10, 1, 1, 55, 55, 1, 1, 220, 1210, 220, 1, 1, 715, 15730, 15730, 715, 1, 1, 2002, 143143, 572572, 143143, 2002, 1, 1, 5005, 1002001, 13026013, 13026013, 1002001, 5005, 1, 1, 11440, 5725720, 208416208, 677352676, 208416208, 5725720, 11440, 1

Offset: 0

Author

Roger L. Bagula, Mar 08 2010

Keywords

• nonn
• tabl

Comments

Triangle of generalized binomial coefficients (n,k)A342889.%20-%20_N.%20J.%20A.%20Sloane">9; cf. A342889. - _N. J. A. Sloane, Apr 03 2021

Row sums are: {1, 2, 12, 112, 1652, 32892, 862864, 28066040, 1105659414, 51177188350, 2734044648194, ...}.

These sequences (q >= 2) are a generalization of A056939.

Examples

			Triangle begins as:
  1.
  1,     1.
  1,    10,       1.
  1,    55,      55,         1.
  1,   220,    1210,       220,         1.
  1,   715,   15730,     15730,       715,         1.
  1,  2002,  143143,    572572,    143143,      2002,       1.
  1,  5005, 1002001,  13026013,  13026013,   1002001,    5005,     1.
  1, 11440, 5725720, 208416208, 677352676, 208416208, 5725720, 11440, 1.
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, 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. A056939 (q=2), A056940 (q=3), A056941 (q=4), A142465 (q=5), A142467 (q=6), A142468 (q=7), this sequence (q=8).

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

Mathematica

c[n_, q_]:= Product[i+j, {j, 0, q}, {i, 1, n}];
T[n_, m_, q_] = c[n, q]/(c[m, q]*c[n - m, q]);
Table[T[n, k, 8], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Apr 13 2019 *)
T[n_, k_, q_]:= (BarnesG[n+q+2]*BarnesG[k+1]*BarnesG[n-k+1]*BarnesG[q+2] )/(BarnesG[n-k+q+2]*BarnesG[k+q+2]*BarnesG[n+1]);
Table[T[n, k, 8], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 13 2019 *)

PARI

{c(m,q) = prod(j=1,m, (j+q)!/(j-1)!)};
for(n=0,10, for(k=0,n, print1(c(n, 8)/(c(k, 8)*c(n-k, 8)), ", "))) \\ G. C. Greubel, Apr 13 2019

Formula

T(n, k, q) = c(n, q)/(c(k, q)*c(n-k, q)) where c(n, q) = Product_{j=1..n} (j+q)!/(j-1)! and q=8.

T(n, k, q) = (G(q+2)*G(k+1)*G(n+q+2)*G(n-k+1))/(G(n+1)*G(k+q+2)*G(n-k+q+ 2)), where G(x) is the Barnes G-function (see A000178). - G. C. Greubel, Apr 13 2019

Extensions

Edited by G. C. Greubel, Apr 13 2019