OFFSET
0,4
FORMULA
a(n) = Sum_{k=1..n} (binomial(n-k,k-1)*Sum_{i=0..n-k} 2^i*binomial(k,n-k-i)* binomial(k+i-1,k-1)*(-1)^(n-k-i)/k).
D-finite with recurrence: (n+1)*a(n) +2*(-2*n+1)*a(n-1) +2*(-n+2)*a(n-2) +2*(2*n-7)*a(n-3) +(n-5)*a(n-4)=0. - R. J. Mathar, Jan 25 2020
From Emanuele Munarini, Jul 11 2024: (Start)
a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n+k-1,3*k)*2^k*Catalan(k).
a(n) ~ (1/2)*sqrt(sqrt(5)/(2*Pi))*(2+sqrt(5))^n/n^(3/2). (End)
a(n) = hypergeom([1/2, 1/2 - n/2, 1 - n/2, n], [1/3, 2/3, 2], 32/27) for n > 0. -Peter Luschny, Jul 11 2024
MATHEMATICA
CoefficientList[Series[-1-(Sqrt[x^4+4*x^3-2*x^2-4*x+1]-x^2-2*x-1)/(4*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Nov 23 2014 *)
a[0] := 0;
a[n_] := HypergeometricPFQ[{1/2, (1 - n)/2, 1 - n/2, n}, {1/3, 2/3, 2}, 32/27];
Table[a[n], {n, 0, 26}] (* Peter Luschny, Jul 11 2024 *)
PROG
(Maxima) a(n):=sum((binomial(n-k, k-1)*sum(2^i*binomial(k, n-k-i)*binomial(k+i-1, k-1)*(-1)^(n-k-i), i, 0, n-k))/k, k, 1, n);
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, Nov 22 2014
STATUS
approved