A258662 - OEIS

%I #10 Jun 14 2015 12:31:41

%S 1,1,4,40,760,23200,1038400,64081600,5214880000,541085248000,

%T 69718686400000,10921720817920000,2044231370959360000,

%U 450550323286412800000,115495483535461427200000,34070943029324134912000000,11460293146666575236608000000,4360020024970859812710400000000,1862768688935303816870072320000000

%N E.g.f. A(x) satisfies: A(x) = exp( Integral A(x) * Integral 1/A(x)^2 dx dx ).

%H Vaclav Kotesovec, <a href="/A258662/b258662.txt">Table of n, a(n) for n = 0..150</a>

%F E.g.f.: 1 / [ Sum_{n>=0} (-1)^n * A063902(n) * x^(2*n) / (2*n)! ].

%F From _Vaclav Kotesovec_, Jun 14 2015: (Start)

%F a(n) ~ c * d^n * n!^2 / sqrt(n), where d = 3*A258895 = 96*Pi / (Gamma(1/6) * Gamma(1/3))^2 = 2^(17/3) * Pi^2 / Gamma(1/3)^6 = 1.3563128975502615865238..., c = 0.8047308283353177558313... = 12/(Gamma(1/3)*Gamma(1/6)) = 2^(7/3)*sqrt(3*Pi) / Gamma(1/3)^3.

%F a(n) ~ 2^(5*n+3) * 3^(n+1) * Pi^(n+1) * n^(2*n+1/2) / (exp(2*n) * Gamma(1/6)^(2*n+1) * Gamma(1/3)^(2*n+1)).

%F a(n) ~ sqrt(3) * 2^((17*n+10)/3) * Pi^(2*n+3/2) * n^(2*n+1/2) / (exp(2*n) * Gamma(1/3)^(6*n+3)).

%F (End)

%e E.g.f.: A(x) = 1 + x^2/2! + 4*x^4/4! + 40*x^6/6! + 760*x^8/8! + 23200*x^10/10! +...

%o (PARI) {a(n) = local(A=1+x); for(i=1, n, A = exp( intformal( A * intformal(1/A^2 + x*O(x^n)) ) ) ); n!*polcoeff(A, n)}

%o for(n=0, 20, print1(a(2*n), ", "))

%Y Cf. A063902, A258895.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jun 06 2015