A110520 Expansion of 1/(1-2*x*c(3*x)), c(x) the g.f. of A000108.

1, 2, 10, 68, 538, 4652, 42628, 406856, 4001914, 40285724, 413049580, 4298523704, 45288486436, 482122686008, 5178044596168, 56038403289488, 610508962548538, 6690154684006268, 73693477140179548, 815508203755227608

Offset

0,2

Comments

Row sums of number triangle A110519.

Hankel transform is A135397. Hankel transform of the aerated sequence is A083667. - Paul Barry, Sep 15 2009

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

J. Abate and W. Whitt, Brownian Motion and the Generalized Catalan Numbers, J. Int. Seq. 14 (2011) # 11.2.6, example section 3.

Formula

a(0)=1, a(n) = Sum_{k=0..n} Sum_{j=0..n} j*C(2n-j-1, n-j)*C(j, k)*3^(n-j)/n, n > 0.

a(n) = Sum_{k=0..n} A039599(n,k)*(-1)^k*3^(n-k). - Philippe Deléham, Dec 11 2007

a(n) = Sum_{k=0..n} A094385(n,k)*2^k. - Philippe Deléham, Feb 26 2009

From Gary W. Adamson, Jul 12 2011: (Start)

a(n) = the top left term in M^n, M = the infinite square production matrix:

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

3, 3, 3, 0, 0, 0, ...

3, 3, 3, 3, 0, 0, ...

3, 3, 3, 3, 3, 0, ...

3, 3, 3, 3, 3, 3, ...

... (End)

n*a(n) + 2*(9-4*n)*a(n-1) + 24*(3-2*n)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011

a(n) ~ 3*12^n/(8*sqrt(Pi)n^(3/2)). - Vaclav Kotesovec, Oct 18 2012

Mathematica

Flatten[{1, Table[Sum[Sum[j*Binomial[2n-j-1, n-j]*Binomial[j, k]*3^(n-j)/n, {j, 0, n}], {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Oct 18 2012 *)

Crossrefs

Cf. A000108, A039599, A094385.

Sequence in context: A152621 A147725 A074603 * A136633 A335729 A361769

Adjacent sequences: A110517 A110518 A110519 * A110521 A110522 A110523

Keyword

easy,nonn

Author

Paul Barry, Jul 24 2005

Status

approved