A157328 Expansion of 1/(1-2x*c(4x)) with c(x) g.f. of Catalan numbers (A000108).

1, 2, 12, 104, 1072, 12192, 147648, 1867392, 24380160, 326105600, 4445965312, 61555599360, 863154221056, 12233140576256, 174954419109888, 2521749245558784, 36595543723671552, 534249057803698176

Offset: 0

Author

Philippe Deléham, Feb 27 2009

Keywords

• nonn

Comments

Hankel transform is A122067.

Crossrefs

Cf. A000108, A000079, A000984, A039599, A064062, A110520, A122067, A151374.

Formula

a(n) = 2^n*A064062(n).

From Paul Barry, Sep 15 2009: (Start)

a(n) = Sum_{k, 0<=k<=n} A039599(n,k)*(-2)^k*4^(n-k).

Integral representation: a(n) = (1/(2*Pi))*Integral(x^n*sqrt(x(16-x))/(x(2+x)),x,0,16). (End)

a(n) = upper left term in M^n, M = an infinite square production matrix as follows:

2, 2, 0, 0, 0, 0, ...

4, 4, 4, 0, 0, 0, ...

4, 4, 4, 4, 0, 0, ...

4, 4, 4, 4, 4, 0, ...

4, 4, 4, 4, 4, 4, ...

...

- Gary W. Adamson, Jul 13 2011

Conjecture: n*a(n) +2*(12-7n)*a(n-1) +16*(3-2n)*a(n-2) = 0. - R. J. Mathar, Dec 14 2011

a(n) = (12*(-1)^n*2^(n - 1)*sqrt(Pi)*n! + 16^n*gamma(n - 1/2)*hypergeometric2F1([1, -n], [3/2 - n], -1/8))/(4*sqrt(Pi)*n!). - Karol A. Penson, Feb 04 2025

Extensions

Entries corrected by R. J. Mathar, Dec 14 2011