OFFSET
1,1
FORMULA
G.f. A(x) = G'(x)*(x*G(x)-x^2)/G(x)^2, where G(x) = A007317(x) = (sqrt(5*x^2-6*x+1)+x-1)/(2*x-2).
a(n) = [x^n] (F(x)^n-F(x)^(n-1)), where F(x) = (x^2-x-1)/(x-1).
a(n) = sum(k=1..n, binomial(n-1,n-k)*sum(i=0..n-k, binomial(k,n-k-i)*binomial(k+i-1,k-1)*2^(-n+2*k+i)*(-1)^(n-k-i))), n>0.
Conjecture D-finite with recurrence: (-n+1)*a(n) +(7*n-11)*a(n-1) +(-11*n+25)*a(n-2) +5*(n-3)*a(n-3)=0. - R. J. Mathar, Oct 07 2016
a(n) ~ 3 * 5^(n - 1/2) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Nov 19 2021
PROG
(Maxima)
a(n):=sum(binomial(n-1, n-k)*sum(binomial(k, n-k-i)*binomial(k+i-1, k-1)*2^(-n+2*k+i)*(-1)^(n-k-i), i, 0, n-k), k, 1, n);
(PARI) x='x+O('x^66); G=(sqrt(5*x^2-6*x+1)+x-1)/(2*x-2); Vec(G' * (x * G - x^2 ) / G^2) \\ Joerg Arndt, Mar 12 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, Mar 12 2014
STATUS
approved