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%I #13 Nov 22 2017 05:27:48
%S 1,1,1,5,15,55,220,876,3645,15485,66735,292155,1293456,5782320,
%T 26071435,118402495,541150155,2487204315,11488482130,53302256250,
%U 248293549685,1160794446445,5444674773325,25614768620105,120837493137460
%N G.f. satisfies A(x) = 1 + x + x^2*A(x)^5.
%H G. C. Greubel, <a href="/A137959/b137959.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{k=0..n-1} C(n-k,k)/(n-k) * C(5*k,n-k-1) for n>0 with a(0)=1. - _Paul D. Hanna_, Jun 16 2009
%F Recurrence: 64*(n-4)*(n-3)*(n-2)*(n-1)*n*(2*n-5)*(2*n-3)*(2*n-1)*(2*n+1)*a(n) = + 5*(n-4)*(n-3)*(n-2)*(2*n-5)*(2*n-3)*(5*n-8)*(5*n-6)*(5*n-4)*(5*n-2)*a(n-2) + 5*(n-4)*(n-3)*(2*n-5)*(5000*n^6 - 45000*n^5 + 157250*n^4 - 267750*n^3 + 227216*n^2 - 87057*n + 11520)*a(n-3) + 15*(n-4)*(5000*n^8 - 80000*n^7 + 532250*n^6 - 1903250*n^5 + 3938648*n^4 - 4710638*n^3 + 3044313*n^2 - 895443*n + 80640)*a(n-4) + 5*(n-2)*(2*n-1)*(5000*n^7 - 95000*n^6 + 734250*n^5 - 2951750*n^4 + 6510194*n^3 - 7505289*n^2 + 3655107*n - 207360)*a(n-5) + 5*(n-3)*(n-2)*n*(2*n-3)*(2*n-1)*(5*n-29)*(5*n-23)*(5*n-17)*(5*n-11)*a(n-6). - _Vaclav Kotesovec_, Sep 18 2013
%F a(n) ~ sqrt(s*(1-s)*(5-6*s) / ((40*s - 40)*Pi)) / (n^(3/2) * r^n), where r = 0.1990700277700792324868112833575428736312653553870... and s = 1.498837534712599040608514104196928592039081694233... are real roots of the system of equations s = 1 + r*(1 + r*s^5), 5 * r^2 * s^4 = 1. - _Vaclav Kotesovec_, Nov 22 2017
%t Flatten[{1,Table[Sum[Binomial[n-k,k]/(n-k)*Binomial[5*k,n-k-1],{k,0,n-1}],{n,1,20}]}] (* _Vaclav Kotesovec_, Sep 18 2013 *)
%o (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^5));polcoeff(A,n)}
%o (PARI) a(n)=if(n==0,1,sum(k=0,n-1,binomial(n-k,k)/(n-k)*binomial(5*k,n-k-1))) \\ _Paul D. Hanna_, Jun 16 2009
%Y Cf. A137960, A137958; A019497, A137954, A137966.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Feb 26 2008
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