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%I #20 Jun 18 2021 10:01:36
%S 1,96,1161216,111588212736,41521527606214656,42355944224989145726976,
%T 96575619003620851215495069696,429963927063544377213100737813282816,
%U 3395036444630744502734855883444511190286336,44244440869926546911112904419213680504885798240256
%N E.g.f. satisfies: A(x) = x + 4*A(x)^5/5.
%C A quadrisection of A249787.
%F E.g.f.: Series_Reversion(x - 4*x^5/5).
%F E.g.f.: Sum_{n>=0} x^(4*n+1)/(4*n+1)! * Product_{k=0..n-1} 4*(5*k+4)!/(5*k)!.
%F a(n) = Product_{k=0..n-1} 4*(5*k+4)!/(5*k)!.
%F a(n) = A249787(4*n+1).
%F For n>0, a(n) = 4^n * (5*n-1)! / ((n-1)! * 5^(n-1)). - _Vaclav Kotesovec_, Nov 15 2014
%F a(n) ~ 2^(2*n) * 5^(4*n+1/2) * n^(4*n) / exp(4*n). - _Vaclav Kotesovec_, Nov 15 2014
%F Recurrence: a(n)+(-2500*n^4+5000*n^3-3500*n^2+1000*n-96)*a(n-1) = 0, a(0) = 1. - _Georg Fischer_, May 29 2021
%e E.g.f.: A(x) = x + 96*x^5/5! + 1161216*x^9/9! + 111588212736*x^13/13! + 41521527606214656*x^17/17! + 42355944224989145726976*x^21/21! +...+ (4/5)^n * (5*n)!/n! * x^(4*n+1)/(4*n+1)! +...
%p rec:={a(n)+(sum([-96,1000,-3500,5000,-2500][i+1]*n^i,i=0..4)*a(n-1))=0,a(0)=1}; f:= gfun:-rectoproc(rec, a(n), remember): map(f, [$0..10]); # _Georg Fischer_, May 29 2021
%t Join[{1}, Table[4^n*(5*n-1)!/((n-1)!*5^(n-1)),{n,1,10}]] (* _Vaclav Kotesovec_, Nov 15 2014 *)
%o (PARI) {a(n)=local(A,X=x+x^2*O(x^n)); A=serreverse(X - 4*x^5/5); n!*polcoeff(A, n)}
%o for(n=0, 15, print1(a(4*n+1), ", "))
%Y Cf. A249787.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 14 2014
%E Offset changed from 1 to 0 by _Georg Fischer_, May 29 2021
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