A137955 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^4)^2.

1, 1, 2, 9, 36, 172, 842, 4310, 22676, 121896, 666884, 3699973, 20771096, 117765084, 673367034, 3878538930, 22483446152, 131070712924, 767929882240, 4519387797894, 26704456819984, 158367557278412, 942285096541344, 5623496055739052, 33653373190735484

Offset: 0

Author

Paul D. Hanna, Feb 26 2008

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Vaclav Kotesovec, Recurrence

Crossrefs

Cf. A137956, A137954; A137952, A137960, A137967.

Programs

Mathematica

Flatten[{1,Table[Sum[Binomial[2*(n-k),k]/(n-k)*Binomial[4*k,n-k-1],{k,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec, Sep 18 2013 *)

PARI

{a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^4)^2);polcoeff(A,n)}

PARI

a(n)=if(n==0,1,sum(k=0,n-1,binomial(2*(n-k),k)/(n-k)*binomial(4*k,n-k-1))) \\ Paul D. Hanna, Jun 16 2009

Formula

G.f.: A(x) = 1 + x*B(x)^2 where B(x) is the g.f. of A137956.

a(n) = Sum_{k=0..n-1} C(2*(n-k),k)/(n-k) * C(4*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009

a(n) ~ sqrt(2*s*(1-s)*(4-5*s) / ((56*s - 48)*Pi)) / (n^(3/2) * r^n), where r = 0.1569043698639381952962655091205241634381480571697... and s = 1.444765371242615455251538467189577278901629278244... are real roots of the system of equations s = 1 + r*(1 + r*s^4)^2, 8 * r^2 * s^3 * (1 + r*s^4) = 1. - Vaclav Kotesovec, Nov 22 2017