A137960 - OEIS

%I #10 Mar 03 2018 13:53:06

%S 1,1,2,11,50,275,1560,9212,56082,348675,2207120,14171155,92075064,

%T 604266000,3999688050,26670727220,178997024610,1208160130227,

%U 8195828345756,55849242272130,382119958804520,2624041637846210

%N G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^5)^2.

%H Vaclav Kotesovec, <a href="/A137960/b137960.txt">Table of n, a(n) for n = 0..400</a>

%F G.f.: A(x) = 1 + x*B(x)^2 where B(x) is the g.f. of A137961.

%F a(n) = Sum_{k=0..n-1} C(2*(n-k),k)/(n-k) * C(5*k,n-k-1) for n>0 with a(0)=1. - _Paul D. Hanna_, Jun 16 2009

%F a(n) ~ sqrt(2*s*(1-s)*(5-6*s) / ((90*s - 80)*Pi)) / (n^(3/2) * r^n), where r = 0.1354712934479194768810666044866029126617104117352... and s = 1.354660923650925199331121807468321286698258863972... are real roots of the system of equations s = 1 + r*(1 + r*s^5)^2, 10 * r^2 * s^4 * (1 + r*s^5) = 1. - _Vaclav Kotesovec_, Nov 22 2017

%o (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^5)^2);polcoeff(A,n)}

%o (PARI) a(n)=if(n==0,1,sum(k=0,n-1,binomial(2*(n-k),k)/(n-k)*binomial(5*k,n-k-1))) \\ _Paul D. Hanna_, Jun 16 2009

%Y Cf. A137961, A137959; A137952, A137955, A137967.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 26 2008