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%I #15 Mar 11 2023 16:55:11
%S 1,5,37,357,3941,46949,587621,7616357,101332837,1375876965,
%T 18987759461,265554114405,3755416368997,53610591434597,
%U 771525112379237,11181285666076517,163041321978836837,2390321854565988197
%N Partial sum of (Catalan numbers A000108 multiplied by powers of 4).
%H Vincenzo Librandi, <a href="/A112698/b112698.txt">Table of n, a(n) for n = 0..300</a>
%F a(n) = Sum_{k=0,..,n} C(k)*4^k, n>=0, with C(n):=A000108(n).
%F G.f.: c(4*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.
%F Recurrence: (n+1)*a(n) = (17*n-7)*a(n-1) - 8*(2*n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 19 2012
%F a(n) ~ 16^(n+1)/(15*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 19 2012
%t CoefficientList[Series[(1-Sqrt[1-16*x])/(8*x)/(1-x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 19 2012 *)
%t With[{nn=20},Accumulate[4^Range[0,nn] CatalanNumber[Range[0,nn]]]] (* _Harvey P. Dale_, Mar 11 2023 *)
%o (PARI) x='x+O('x^50); Vec((1-sqrt(1-16*x))/(8*x*(1-x))) \\ _G. C. Greubel_, Mar 17 2017
%Y Fifth column (m=4) of triangle A112705.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Oct 31 2005
%E Definition clarified by _Harvey P. Dale_, Mar 11 2023
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