A142468 An eight-products triangle sequence of coefficients: T(n,k) = binomial(n,k) * Product_{j=1..7} j!*(n+j)!/((k+j)!*(n-k+j)!).

1, 1, 1, 1, 9, 1, 1, 45, 45, 1, 1, 165, 825, 165, 1, 1, 495, 9075, 9075, 495, 1, 1, 1287, 70785, 259545, 70785, 1287, 1, 1, 3003, 429429, 4723719, 4723719, 429429, 3003, 1, 1, 6435, 2147145, 61408347, 184225041, 61408347, 2147145, 6435, 1

Offset: 0

Author

Roger L. Bagula, Sep 20 2008

Keywords

• nonn
• tabl

Comments

Triangle of generalized binomial coefficients (n,k)A342889.%20-%20_N.%20J.%20A.%20Sloane">8; cf. A342889. - _N. J. A. Sloane, Apr 03 2021

Row sums are {1, 2, 11, 92, 1157, 19142, 403691, 10312304, 311348897, 10826298914, 426196716090, ...}.

The general function is T(n,m)L = binomial(n,m)*Product{k=1..L} k!*(n + k)!/((m + k)!*(n - m + k)!) to give the quadratic row {1, L+2, 1}.

Examples

			Triangle begins as:
  1;
  1,    1;
  1,    9,       1;
  1,   45,      45,        1;
  1,  165,     825,      165,         1;
  1,  495,    9075,     9075,       495,        1;
  1, 1287,   70785,   259545,     70785,     1287,       1;
  1, 3003,  429429,  4723719,   4723719,   429429,    3003,    1;
  1, 6435, 2147145, 61408347, 184225041, 61408347, 2147145, 6435, 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. A001263, 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

B:=Binomial; [B(n,k)*(&*[B(n+2*j, k+j)/B(n+2*j, j): j in [1..7]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 14 2019, Mar 03 2021

Maple

b:= binomial;
T:= (n, k) -> b(n, k)*mul(b(n+2*j, k+j)/b(n+2*j, j), j = 1..7);
seq(seq(T(n, k), k = 0..n), n = 0..10); # G. C. Greubel, Nov 14 2019, Mar 03 2021

Mathematica

T[n_, k_]:= T[n,k]= With[{B=Binomial}, B[n,k]* Product[B[n+2*j,k+j]/B[n+2*j,j], {j, 7}] ];
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Nov 14 2019, Mar 03 2021 *)

PARI

T(n,k) = b=binomial; b(n,k)*prod(j=1,7, b(n+ 2*j,k+j)/b(n+2*j,j)); \\ G. C. Greubel, Nov 14 2019, Mar 03 2021

Sage

b=binomial;
def T(n, k): return b(n,k)*product(b(n+2*j,k+j)/b(n+2*j,j) for j in (1..7))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 14 2019, Mar 03 2021

Formula

T(n,k) = binomial(n,k)*Product_{j=1..7} j!*(n+j)!/((k+j)!*(n-k+j)!).

Extensions

Edited by G. C. Greubel, Nov 14 2019