|
%I #25 Sep 08 2022 08:46:00
%S 1,1,315,975975,12909521625,495181420358625,44035787449951171875,
%T 7845481113748784765634375,2526730187976408357560632640625,
%U 1362965093449949100037985665872890625,1160978904909328561005478318639484556796875
%N E.g.f. satisfies: A(x) = exp(x^4*A(x)^4/4!).
%H G. C. Greubel, <a href="/A200314/b200314.txt">Table of n, a(n) for n = 0..100</a>
%F a(n) = (4*n+1)^(n-1) * (4*n)!/(n!*(4!)^n).
%F E.g.f.: (1/x)*Series_Reversion( x*exp(-x^4/4!) ).
%F Powers of e.g.f.: define a(n,m) by A(x)^m = Sum_{n>=0} a(n,m)*x^(4*n)/(4*n)!
%F then a(n,m) = m*(4*n+m)^(n-1) * (4*n)!/(n!*(4!)^n).
%e E.g.f.: A(x) = 1 + x^4/4! + 315*x^8/8! + 975975*x^12/12! + ...
%e where log(A(x)) = x^4*A(x)^4/4! and
%e A(x)^4 = 1 + 4*x^4/4! + 1680*x^8/8! + 5913600*x^12/12! + 84084000000*x^16/16! + ...
%t Table[(4*n + 1)^(n - 1)*(4*n)!/(n!*(4!)^n), {n, 0, 30}] (* _G. C. Greubel_, Jul 27 2018 *)
%o (PARI) {a(n)=(4*n)!*polcoeff(1/x*serreverse(x*(exp(-x^4/4!+x*O(x^(4*n))))),4*n)}
%o (PARI) {a(n)=(4*n+1)^(n-1)*(4*n)!/(n!*(4!)^n)};
%o (Magma) [(4*n+1)^(n-1)*Factorial(4*n)/(24^n*Factorial(n)): n in [0..30]]; // _G. C. Greubel_, Jul 27 2018
%o (GAP) List([0..10],n->(4*n+1)^(n-1)*Factorial(4*n)/(Factorial(n)*Factorial(4)^n)); # _Muniru A Asiru_, Jul 28 2018
%Y Cf. A034941, A200313, A200315.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 15 2011
| 