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

%I A137954 #17 May 04 2024 02:49:46
%S A137954 1,1,1,4,10,32,107,360,1270,4544,16537,61092,228084,860056,3269994,
%T A137954 12521488,48250690,186959312,727989318,2847167632,11179394088,
%U A137954 44053232012,174160578150,690576010820,2745713062854,10944253432600
%N A137954 G.f. satisfies A(x) = 1 + x + x^2*A(x)^4.
%H A137954 G. C. Greubel, <a href="/A137954/b137954.txt">Table of n, a(n) for n = 0..1000</a>
%F A137954 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
%F A137954 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
%F A137954 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
%t A137954 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 *)
%o A137954 (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x+x^2*A^4);polcoeff(A,n)}
%o A137954 (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
%Y A137954 Cf. A137955, A137953; A019497, A137959, A137966.
%K A137954 nonn
%O A137954 0,4
%A A137954 _Paul D. Hanna_, Feb 26 2008