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

0,5

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.

Formula

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

Example

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;
  ...

Prog

(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));

Crossrefs

Cf. A007318, A363838.

Sequence in context: A129439 A176469 A141542 * A129453 A129455 A329322

Adjacent sequences: A364809 A364810 A364811 * A364813 A364814 A364815

Keyword

nonn,tabl

Author

Michel Marcus, Oct 21 2023

Status

approved