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A387244 Expansion of e.g.f. exp(x^2/(1-x)^4).

%I A387244 #16 Aug 25 2025 04:41:57
%S A387244 1,0,2,24,252,2880,38280,594720,10565520,209502720,4558407840,
%T A387244 107702179200,2744400415680,75016089308160,2189152249764480,
%U A387244 67906418407027200,2230160988344889600,77271779968704921600,2815893910009609228800,107629691727791474841600,4304364116456244429388800
%N A387244 Expansion of e.g.f. exp(x^2/(1-x)^4).
%C A387244 In general, if s >= 1, 1 <= r <= s and e.g.f. = exp(x^r/(1-x)^s) then for n > 0, a(n) = n! * Sum_{k=1..n} binomial(n + (s-r)*k - 1, s*k - 1)/k!.
%H A387244 Vincenzo Librandi, <a href="/A387244/b387244.txt">Table of n, a(n) for n = 0..200</a>
%F A387244 For n > 0, a(n) = n! * Sum_{k=1..n} binomial(n + 2*k - 1, 4*k - 1)/k!.
%F A387244 a(n) = 5*(n-1)*a(n-1) - 2*(n-1)*(5*n-11)*a(n-2) + 2*(n-2)*(n-1)*(5*n-14)*a(n-3) - 5*(n-4)*(n-3)*(n-2)*(n-1)*a(n-4) + (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).
%F A387244 a(n) ~ 2^(1/5) * 5^(-1/2) * exp(1/80 - 2^(-9/5)*n^(2/5)/3 + 5*2^(-8/5)*n^(4/5) - n) * n^(n - 1/10).
%t A387244 nmax=20; CoefficientList[Series[E^(x^2/(1-x)^4), {x, 0, nmax}], x] * Range[0, nmax]!
%t A387244 nmax=20; Join[{1}, Table[n!*Sum[Binomial[n+2*k-1, 4*k-1]/k!, {k, 1, n}], {n, 1, nmax}]]
%t A387244 Join[{1}, Table[n!*n*(n - 1)*(n + 1)/6 * HypergeometricPFQ[{1 - n/2, 3/2 - n/2, 1 + n/2, 3/2 + n/2}, {5/4, 3/2, 7/4, 2}, 1/16], {n, 1, 20}]]
%o A387244 (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x^2/(1-x)^4))); [Factorial(n-1)*b[n]: n in [1..m]]; // _Vincenzo Librandi_, Aug 25 2025
%Y A387244 Cf. A293012, A000262, A082579, A091695, A361283.
%Y A387244 Cf. A052845, A052887, A386514.
%K A387244 nonn,new
%O A387244 0,3
%A A387244 _Vaclav Kotesovec_, Aug 24 2025