A200314 E.g.f. satisfies: A(x) = exp(x^4*A(x)^4/4!).

1, 1, 315, 975975, 12909521625, 495181420358625, 44035787449951171875, 7845481113748784765634375, 2526730187976408357560632640625, 1362965093449949100037985665872890625, 1160978904909328561005478318639484556796875

Offset: 0

Author

Paul D. Hanna, Nov 15 2011

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x^4/4! + 315*x^8/8! + 975975*x^12/12! + ...
where log(A(x)) = x^4*A(x)^4/4! and
A(x)^4 = 1 + 4*x^4/4! + 1680*x^8/8! + 5913600*x^12/12! + 84084000000*x^16/16! + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A034941, A200313, A200315.

Programs

GAP

List([0..10],n->(4*n+1)^(n-1)*Factorial(4*n)/(Factorial(n)*Factorial(4)^n)); # Muniru A Asiru, Jul 28 2018

Magma

[(4*n+1)^(n-1)*Factorial(4*n)/(24^n*Factorial(n)): n in [0..30]]; // G. C. Greubel, Jul 27 2018

Mathematica

Table[(4*n + 1)^(n - 1)*(4*n)!/(n!*(4!)^n), {n, 0, 30}] (* G. C. Greubel, Jul 27 2018 *)

PARI

{a(n)=(4*n)!*polcoeff(1/x*serreverse(x*(exp(-x^4/4!+x*O(x^(4*n))))),4*n)}

PARI

{a(n)=(4*n+1)^(n-1)*(4*n)!/(n!*(4!)^n)};

Formula

a(n) = (4*n+1)^(n-1) * (4*n)!/(n!*(4!)^n).

E.g.f.: (1/x)*Series_Reversion( x*exp(-x^4/4!) ).

Powers of e.g.f.: define a(n,m) by A(x)^m = Sum_{n>=0} a(n,m)*x^(4*n)/(4*n)!

then a(n,m) = m*(4*n+m)^(n-1) * (4*n)!/(n!*(4!)^n).