A137956 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^2)^4.

1, 1, 4, 14, 64, 301, 1500, 7738, 40948, 221278, 1215284, 6765148, 38083556, 216431253, 1240048740, 7155236960, 41542685352, 242513393884, 1422608044604, 8381507029660, 49574494112992, 294260899150492, 1752288415205896

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Vaclav Kotesovec, Recurrence

Formula

G.f.: A(x) = 1 + x*B(x)^4 where B(x) is the g.f. of A137955.

a(n) = Sum_{k=0..n-1} C(4*(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(4*s*(1-s)*(2-3*s) / ((28*s - 16)*Pi)) / (n^(3/2) * r^n), where r = 0.1569043698639381952962655091205241634381480571697... and s = 1.683635070625292013962854364673077567156937629734... are real roots of the system of equations s = 1 + r*(1 + r*s^2)^4, 8 * r^2 * s * (1 + r*s^2)^3 = 1. - Vaclav Kotesovec, Nov 22 2017

Mathematica

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

Prog

(PARI) {a(n)=local(A=1+x*O(x^n)); for(i=0, n, A=1+x*(1+x*A^2)^4); polcoeff(A, n)}
(PARI) a(n)=if(n==0, 1, sum(k=0, n-1, binomial(4*(n-k), k)/(n-k)*binomial(2*k, n-k-1))) \\ Paul D. Hanna, Jun 16 2009

Crossrefs

Cf. A137955, A137957; A137958, A137964, A137971.

Sequence in context: A149494 A322206 A149495 * A357793 A341682 A242764

Adjacent sequences: A137953 A137954 A137955 * A137957 A137958 A137959

Keyword

nonn

Author

Paul D. Hanna, Feb 26 2008

Status

approved