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%I #38 Sep 08 2022 08:46:23
%S 1,1,6,31,176,1071,6797,44493,298279,2037550,14131441,99244564,
%T 704360703,5043969503,36399930179,264451303466,1932650461883,
%U 14198082537190,104792195449688,776681663951998,5778226417888171,43135097969972931,323012620411650708,2425745980876575899,18264470545275495152
%N G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^5).
%H Seiichi Manyama, <a href="/A321799/b321799.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..199 from Ludovic Schwob)
%F a(n) = Sum_{k=0..n} (C(n,k)/(n-k+1)) * C(n+4*k-1,n-k).
%F a(n) ~ sqrt((1 - r*s)*(1 + 4*r*s) / (5*Pi*(5*s - 2))) / (2 * n^(3/2) * r^(n+1)), where r = 0.124910212976238209867004924637837518925706044646... and s = 1.72708330560542094133450070142549940430523638921... are real roots of the system of equations s*(1 - r/(1 - r*s)^5) = 1, 5*r^2*s^2 = (1 - r*s)^6. - _Vaclav Kotesovec_, Nov 21 2018
%p eq:= a - 1/(1-x/(1-x*a)^5):
%p S:= series(RootOf(numer(eq),a),x,31):
%p seq(coeff(S,x,j),j=0..30); # _Robert Israel_, Dec 10 2018
%t a[n_]:=Sum[ Binomial[n,k]/(n-k+1)*Binomial[n+4*k-1,n-k], {k,0,n}]; Array[a, 20, 0] (* _Stefano Spezia_, Nov 19 2018 *)
%t A[_] = 0; Do[A[x_] = 1/(1-x/(1-x*A[x])^5)+O[x]^25, {25}];
%t CoefficientList[A[x], x] (* _Jean-François Alcover_, Dec 30 2018 *)
%o (PARI) a(n) = sum(k=0, n, binomial(n,k)*binomial(n+4*k-1, n-k)/(n-k+1)); \\ _Michel Marcus_, Nov 19 2018
%o (GAP) List([0..25],n->Sum([0..n],k->Binomial(n,k)/(n-k+1)*Binomial(n+4*k-1,n-k))); # _Muniru A Asiru_, Nov 24 2018
%o (Magma) [1] cat [&+[(Binomial(n,k)/(n-k+1)) * Binomial(n+4*k-1,n-k): k in [0..n]]: n in [1.. 25]]; // _Vincenzo Librandi_, Dec 08 2018
%o (Sage) [sum(binomial(n,k)*binomial(n+4*k-1,n-k)/(n-k+1) for k in (0..n)) for n in range(25)] # _G. C. Greubel_, Dec 14 2018
%Y Cf. A109081, A161797, A321798.
%K nonn
%O 0,3
%A _Ludovic Schwob_, Nov 19 2018
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