A137971 - OEIS

%I #10 Mar 03 2018 12:28:26

%S 1,1,4,30,232,2037,18720,179454,1770380,17864490,183510672,1912621814,

%T 20175123732,214980182783,2310645275932,25021270486830,

%U 272717638241172,2989549949264304,32938634975109864,364566094737276708

%N G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^6)^4.

%H Vaclav Kotesovec, <a href="/A137971/b137971.txt">Table of n, a(n) for n = 0..350</a>

%F G.f.: A(x) = 1 + x*B(x)^4 where B(x) is the g.f. of A137972.

%F a(n) = Sum_{k=0..n-1} C(4*(n-k),k)/(n-k) * C(6*k,n-k-1) for n>0 with a(0)=1. - _Paul D. Hanna_, Jun 16 2009

%F a(n) ~ sqrt(4*s*(1-s)*(6-7*s) / ((276*s - 240)*Pi)) / (n^(3/2) * r^n), where r = 0.0833821738312503523008482260558417829257343369560... and s = 1.229254439060935443156800948762443928645579909446... are real roots of the system of equations s = 1 + r*(1 + r*s^6)^4, 24 * r^2 * s^5 * (1 + r*s^6)^3 = 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^6)^4);polcoeff(A,n)}

%o (PARI) a(n)=if(n==0,1,sum(k=0,n-1,binomial(4*(n-k),k)/(n-k)*binomial(6*k,n-k-1))) \\ _Paul D. Hanna_, Jun 16 2009

%Y Cf. A137972, A137970; A137956, A137958, A137964.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 26 2008