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

0,2

Comments

Hankel transform is A122067.

Links

Table of n, a(n) for n=0..17.

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

Crossrefs

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

Sequence in context: A050621 A326237 A152254 * A061632 A301833 A320899

Adjacent sequences: A157325 A157326 A157327 * A157329 A157330 A157331

Keyword

nonn

Author

Philippe Deléham, Feb 27 2009

Extensions

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

Status

approved