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A300591 O.g.f. A(x) satisfies: [x^n] exp( n^2 * A(x) ) = n^2 * [x^(n-1)] exp( n^2 * A(x) ) for n>=1.

%I A300591 #19 Oct 13 2020 12:39:02
%S A300591 1,2,27,736,30525,1715454,123198985,10931897664,1172808994833,
%T A300591 149774206572050,22487782439633786,3927856758905547936,
%U A300591 790620718368726490063,181836026214536919343314,47416473117145116482171400,13920906749656695367066255360,4572270908185359745686931830057,1670388578072378805032472463218378,675225859431899136993903503004997481,300576566118865697499246162737030656800
%N A300591 O.g.f. A(x) satisfies: [x^n] exp( n^2 * A(x) ) = n^2 * [x^(n-1)] exp( n^2 * A(x) ) for n>=1.
%C A300591 Compare to: [x^n] exp( n^2 * x ) = n * [x^(n-1)] exp( n^2 * x ) for n>=1.
%C A300591 It is conjectured that this sequence consists entirely of integers.
%C A300591 a(n) is divisible by n (conjecture): A300598(n) = a(n)/n for n>=1.
%H A300591 Paul D. Hanna, <a href="/A300591/b300591.txt">Table of n, a(n) for n = 1..200</a>
%F A300591 O.g.f. equals the logarithm of the e.g.f. of A300590.
%F A300591 a(n) ~ c * n!^2 * n^2, where c = 0.1354708370957778563796... - _Vaclav Kotesovec_, Oct 13 2020
%e A300591 O.g.f.: A(x) = x + 2*x^2 + 27*x^3 + 736*x^4 + 30525*x^5 + 1715454*x^6 + 123198985*x^7 + 10931897664*x^8 + 1172808994833*x^9 + 149774206572050*x^10 + ...
%e A300591 where
%e A300591 exp(A(x)) = 1 + x + 5*x^2/2! + 175*x^3/3! + 18385*x^4/4! + 3759701*x^5/5! + 1258735981*x^6/6! + 630063839035*x^7/7! + 445962163492385*x^8/8! + ... + A300590(n)*x^n/n! + ...
%e A300591 such that: [x^n] exp( n^2 * A(x) ) = n^2 * [x^(n-1)] exp( n^2 * A(x) ).
%o A300591 (PARI) {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)^2)); A[#A] = ((#A-1)^2*V[#A-1] - V[#A])/(#A-1)^2 ); polcoeff( log(Ser(A)), n)}
%o A300591 for(n=1, 30, print1(a(n), ", "))
%Y A300591 Cf. A300590, A300598, A300871, A296171, A300593, A300595.
%K A300591 nonn
%O A300591 1,2
%A A300591 _Paul D. Hanna_, Mar 09 2018