A086619 Product of first n terms of the binomial transform of the Catalan numbers (A007317).

1, 2, 10, 150, 7650, 1438200, 1051324200, 3101406390000, 37945707181650000, 1966422437567466300000, 438887790263120370963300000, 427664112802721593716655529100000

Offset: 0

Author

Paul D. Hanna, Jul 24 2003

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..56

Crossrefs

Cf. A007317, A003046, A086620.

Programs

Mathematica

Table[Product[Sum[Binomial[m, k]*Binomial[2*k, k]/(k+1), {k, 0, m}], {m, 0, n}], {n, 0, 12}] (* Vaclav Kotesovec, Oct 27 2017 *)
Table[Product[Hypergeometric2F1[1/2, -k, 2, -4], {k, 0, n}], {n, 0, 12}] (* Vaclav Kotesovec, Oct 27 2017 *)

Formula

Determinant of the n X n matrix in the upper left corner of square table A086620.

a(n) ~ c * 5^(n*(n+4)/2) * exp(3*n/2) / (2^(3*n) * Pi^(n/2) * n^(3*n/2 + 27/16)), where c = 0.286457028871544332017553737430712054187435533562040177617845850168... - Vaclav Kotesovec, Oct 27 2017