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%I #16 Jan 30 2020 21:29:17
%S 1,1,3,4,11,21,48,106,235,535,1203,2751,6272,14392,33078,76224,176043,
%T 407253,943833,2190397,5090371,11843689,27586793,64320191,150102784,
%U 350586496,819477792,1916861350,4486760870,10508582130,24626700888
%N G.f.: (2*x^2+4*x+3)/((2*x+2)*sqrt(-4*x^3-4*x^2+1))-1/(2*x+2).
%F a(n) = n*sum(k=1..n, (binomial(2*k,n-k)*binomial(n-k-1,k-1))/k, n>0, a(0)=1.
%F D-finite with recurrence: 3*n*a(n) +(7*n-8)*a(n-1) +2*(-3*n-2)*a(n-2) +2*(-19*n+35)*a(n-3) +2*(-26*n+81)*a(n-4) +4*(-8*n+35)*a(n-5) +4*(-2*n+11)*a(n-6)=0. - _R. J. Mathar_, Jan 25 2020
%t CoefficientList[Series[(2 x^2 + 4 x + 3) / ((2 x + 2) Sqrt[-4 x^3 - 4 x^2 + 1]) - 1 / (2 x + 2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Nov 22 2014 *)
%o (Maxima)
%o a(n):=if n=0 then 1 else n*sum((binomial(2*k,n-k)*binomial(n-k-1,k-1))/k,k,1,n);
%Y Cf. A007477.
%K nonn
%O 0,3
%A _Vladimir Kruchinin_, Nov 22 2014
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