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%I #11 Oct 14 2020 02:42:01
%S 1,4,243,40448,12519125,6111917748,4308276119854,4151360558858752,
%T 5268077625693186225,8567999843251994553500,
%U 17491034862909191177473132,44081002571786307016424642880,135294991782833277819666508563622,499980220350805674732349875409752344,2203045016526580123761644939382016407000,11476028442989415865296132639050660100915200
%N O.g.f. A(x) satisfies: [x^n] exp( n^3 * A(x) ) = n^3 * [x^(n-1)] exp( n^3 * A(x) ) for n>=1.
%C Compare to: [x^n] exp( n^3 * x ) = n^2 * [x^(n-1)] exp( n^3 * x ) for n>=1.
%C It is conjectured that this sequence consists entirely of integers.
%H Paul D. Hanna, <a href="/A300595/b300595.txt">Table of n, a(n) for n = 1..200</a>
%F O.g.f. equals the logarithm of the e.g.f. of A300594.
%F a(n) ~ c * n!^3 * n^3, where c = 0.40774346023... - _Vaclav Kotesovec_, Oct 14 2020
%e O.g.f.: A(x) = x + 4*x^2 + 243*x^3 + 40448*x^4 + 12519125*x^5 + 6111917748*x^6 + 4308276119854*x^7 + 4151360558858752*x^8 + 5268077625693186225*x^9 + ...
%e where
%e exp(A(x)) = 1 + x + 9*x^2/2! + 1483*x^3/3! + 976825*x^4/4! + 1507281021*x^5/5! + 4409747597401*x^6/6! + 21744850191313999*x^7/7! + ... + A300594(n)*x^n/n! + ...
%e such that: [x^n] exp( n^3 * A(x) ) = n^3 * [x^(n-1)] exp( n^3 * A(x) ).
%o (PARI) {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)^3)); A[#A] = ((#A-1)^3*V[#A-1] - V[#A])/(#A-1)^3 ); polcoeff( log(Ser(A)), n)}
%o for(n=1, 30, print1(a(n), ", "))
%Y Cf. A300594, A296171, A300591, A300593.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Mar 09 2018
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