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A137961
G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^2)^5.
6
1, 1, 5, 20, 105, 575, 3306, 19760, 121035, 757230, 4815530, 31039402, 202335855, 1331569725, 8834918160, 59035907480, 396937508816, 2683518356850, 18230570856710, 124390574548960, 852074529347120, 5857453791938135
OFFSET
0,3
LINKS
FORMULA
G.f.: A(x) = 1 + x*B(x)^5 where B(x) is the g.f. of A137960.
a(n) = Sum_{k=0..n-1} C(5*(n-k),k)/(n-k) * C(2*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009
a(n) ~ sqrt(5*s*(1-s)*(2-3*s) / ((36*s - 20)*Pi)) / (n^(3/2) * r^n), where r = 0.1354712934479194768810666044866029126617104117352... and s = 1.618016818966151244027202981456410137451426090894... are real roots of the system of equations s = 1 + r*(1 + r*s^2)^5, 10 * r^2 * s * (1 + r*s^2)^4 = 1. - Vaclav Kotesovec, Nov 22 2017
PROG
(PARI) {a(n)=local(A=1+x*O(x^n)); for(i=0, n, A=1+x*(1+x*A^2)^5); polcoeff(A, n)}
(PARI) a(n)=if(n==0, 1, sum(k=0, n-1, binomial(5*(n-k), k)/(n-k)*binomial(2*k, n-k-1))) \\ Paul D. Hanna, Jun 16 2009
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 26 2008
STATUS
approved