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
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Programs
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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 *) -
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
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