A137954 G.f. satisfies A(x) = 1 + x + x^2*A(x)^4.

1, 1, 1, 4, 10, 32, 107, 360, 1270, 4544, 16537, 61092, 228084, 860056, 3269994, 12521488, 48250690, 186959312, 727989318, 2847167632, 11179394088, 44053232012, 174160578150, 690576010820, 2745713062854, 10944253432600

Offset

0,4

Links

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

Formula

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

Recurrence: 3*(n-1)*n*(3*n-8)*(3*n-5)*(3*n-2)*(3*n+2)*a(n) = 64*(n-1)^2*(2*n-3)*(2*n-1)*(3*n-8)*(3*n-5)*a(n-2) + 32*(2*n-3)*(3*n-8)*(36*n^4 - 204*n^3 + 364*n^2 - 216*n + 35)*a(n-3) + 16*(3*n-2)*(144*n^5 - 1536*n^4 + 6005*n^3 - 10278*n^2 + 6790*n - 600)*a(n-4) + 8*n*(2*n-7)*(3*n-5)*(3*n-2)*(4*n-19)*(4*n-9)*a(n-5). - Vaclav Kotesovec, Sep 18 2013

a(n) ~ sqrt(s*(1-s)*(4-5*s) / ((24*s - 24)*Pi)) / (n^(3/2) * r^n), where r = 0.2362629484147719796376166796890824064312524895955... and s = 1.648350597886362639516822239585443208575003319460... are real roots of the system of equations s = 1 + r*(1 + r*s^4), 4 * r^2 * s^3 = 1. - Vaclav Kotesovec, Nov 22 2017

Mathematica

Flatten[{1, Table[Sum[Binomial[n-k, k]/(n-k)*Binomial[4*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+x^2*A^4); polcoeff(A, n)}
(PARI) a(n)=if(n==0, 1, sum(k=0, n-1, binomial(n-k, k)/(n-k)*binomial(4*k, n-k-1))) \\ Paul D. Hanna, Jun 16 2009

Crossrefs

Cf. A137955, A137953; A019497, A137959, A137966.

Sequence in context: A129880 A303832 A316103 * A028283 A196991 A151746

Adjacent sequences: A137951 A137952 A137953 * A137955 A137956 A137957

Keyword

nonn

Author

Paul D. Hanna, Feb 26 2008

Status

approved