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

Author

Paul Barry, Jul 24 2005

Keywords

• easy
• nonn

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.

Crossrefs

Cf. A000108, A039599, A094385.

Programs

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 *)

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