A364812 Triangle of generalized binomial coefficients T(n,k) = ff(n)/(ff(k)*ff(n-k)) where ff(n) = A363838(n), the generalized factorial.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 16, 24, 16, 1, 1, 5, 40, 40, 5, 1, 1, 36, 90, 480, 90, 36, 1, 1, 7, 126, 210, 210, 126, 7, 1, 1, 256, 896, 10752, 3360, 10752, 896, 256, 1, 1, 81, 10368, 24192, 54432, 54432, 24192, 10368, 81, 1, 1, 100, 4050, 345600, 151200, 1088640, 151200, 345600, 4050, 100, 1

Offset: 0

Author

Michel Marcus, Oct 21 2023

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3,   1;
  1, 16, 24,  16,  1;
  1,  5, 40,  40,  5, 1;
  1, 36, 90, 480, 90, 36, 1;
  ...
		

Links

• Michel Marcus, Table of n, a(n) for n = 0..5150 (Rows n=0..100 flattened).
• Jeffrey C. Lagarias and Wijit Yangjit, The factorial function and generalizations, extended, arXiv:2310.12949 [math.NT], 2023. See Table 3 p. 30.

Crossrefs

Cf. A007318, A363838.

Programs

PARI

f(n, b) = sum(i=1, logint(n, b), n\b^i);
ff(n) = prod(b=2, n, b^f(n,b)); \\ A363838
T(n,k) = ff(n)/(ff(k)*ff(n-k));
row(n) = vector(n+1, k, T(n, k-1));

Formula

T(n,k) = A363838(n)/(A363838(k)*A363838(n-k)).