login
Expansion of 1/(1-2*x*c(3*x)), c(x) the g.f. of A000108.
6

%I #27 Jul 08 2021 11:40:42

%S 1,2,10,68,538,4652,42628,406856,4001914,40285724,413049580,

%T 4298523704,45288486436,482122686008,5178044596168,56038403289488,

%U 610508962548538,6690154684006268,73693477140179548,815508203755227608

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

%C Row sums of number triangle A110519.

%C Hankel transform is A135397. Hankel transform of the aerated sequence is A083667. - _Paul Barry_, Sep 15 2009

%H Vincenzo Librandi, <a href="/A110520/b110520.txt">Table of n, a(n) for n = 0..200</a>

%H J. Abate and W. Whitt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Whitt/whitt6.html">Brownian Motion and the Generalized Catalan Numbers</a>, J. Int. Seq. 14 (2011) # 11.2.6, example section 3.

%F 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.

%F a(n) = Sum_{k=0..n} A039599(n,k)*(-1)^k*3^(n-k). - _Philippe Deléham_, Dec 11 2007

%F a(n) = Sum_{k=0..n} A094385(n,k)*2^k. - _Philippe Deléham_, Feb 26 2009

%F From _Gary W. Adamson_, Jul 12 2011: (Start)

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

%F 2, 2, 0, 0, 0, 0, ...

%F 3, 3, 3, 0, 0, 0, ...

%F 3, 3, 3, 3, 0, 0, ...

%F 3, 3, 3, 3, 3, 0, ...

%F 3, 3, 3, 3, 3, 3, ...

%F ... (End)

%F n*a(n) + 2*(9-4*n)*a(n-1) + 24*(3-2*n)*a(n-2) = 0. - _R. J. Mathar_, Nov 14 2011

%F a(n) ~ 3*12^n/(8*sqrt(Pi)n^(3/2)). - _Vaclav Kotesovec_, Oct 18 2012

%t 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 *)

%Y Cf. A000108, A039599, A094385.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Jul 24 2005