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

%I A110520 #27 Jul 08 2021 11:40:42
%S A110520 1,2,10,68,538,4652,42628,406856,4001914,40285724,413049580,
%T A110520 4298523704,45288486436,482122686008,5178044596168,56038403289488,
%U A110520 610508962548538,6690154684006268,73693477140179548,815508203755227608
%N A110520 Expansion of 1/(1-2*x*c(3*x)), c(x) the g.f. of A000108.
%C A110520 Row sums of number triangle A110519.
%C A110520 Hankel transform is A135397. Hankel transform of the aerated sequence is A083667. - _Paul Barry_, Sep 15 2009
%H A110520 Vincenzo Librandi, <a href="/A110520/b110520.txt">Table of n, a(n) for n = 0..200</a>
%H A110520 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 A110520 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 A110520 a(n) = Sum_{k=0..n} A039599(n,k)*(-1)^k*3^(n-k). - _Philippe Deléham_, Dec 11 2007
%F A110520 a(n) = Sum_{k=0..n} A094385(n,k)*2^k. - _Philippe Deléham_, Feb 26 2009
%F A110520 From _Gary W. Adamson_, Jul 12 2011: (Start)
%F A110520 a(n) = the top left term in M^n, M = the infinite square production matrix:
%F A110520   2, 2, 0, 0, 0, 0, ...
%F A110520   3, 3, 3, 0, 0, 0, ...
%F A110520   3, 3, 3, 3, 0, 0, ...
%F A110520   3, 3, 3, 3, 3, 0, ...
%F A110520   3, 3, 3, 3, 3, 3, ...
%F A110520   ... (End)
%F A110520 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 A110520 a(n) ~ 3*12^n/(8*sqrt(Pi)n^(3/2)). - _Vaclav Kotesovec_, Oct 18 2012
%t A110520 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 A110520 Cf. A000108, A039599, A094385.
%K A110520 easy,nonn
%O A110520 0,2
%A A110520 _Paul Barry_, Jul 24 2005