A137968 - OEIS

%I #10 Mar 03 2018 13:52:42

%S 1,1,6,27,158,981,6342,42728,295008,2079882,14908740,108312873,

%T 795836544,5903472999,44151306690,332552305818,2520416719368,

%U 19207222744326,147086508325056,1131292622149352,8735383810590486

%N G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^2)^6.

%H Vaclav Kotesovec, <a href="/A137968/b137968.txt">Table of n, a(n) for n = 0..400</a>

%F G.f.: A(x) = 1 + x*B(x)^6 where B(x) is the g.f. of A137967.

%F a(n) = Sum_{k=0..n-1} C(6*(n-k),k)/(n-k) * C(2*k,n-k-1) for n>0 with a(0)=1. - _Paul D. Hanna_, Jun 16 2009

%F a(n) ~ sqrt(6*s*(1-s)*(2-3*s) / ((44*s - 24)*Pi)) / (n^(3/2) * r^n), where r = 0.1201742080825038015263858974579392344239858277873... and s = 1.572098697306844482137442690518486437859864764710... are real roots of the system of equations s = 1 + r*(1 + r*s^2)^6, 12 * r^2 * s * (1 + r*s^2)^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^2)^6);polcoeff(A,n)}

%o (PARI) a(n)=if(n==0,1,sum(k=0,n-1,binomial(6*(n-k),k)/(n-k)*binomial(2*k,n-k-1))) \\ _Paul D. Hanna_, Jun 16 2009

%Y Cf. A137967, A137969; A137970, A137972, A137974.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 26 2008