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A300593 O.g.f. A(x) satisfies: [x^n] exp( n^2 * A(x) ) = n^3 * [x^(n-1)] exp( n^2 * A(x) ) for n>=1.

%I A300593 #12 Oct 14 2020 02:36:30
%S A300593 1,6,216,18016,2718575,667151244,249904389518,136335045655680,
%T A300593 104258627494173747,108236370325030253850,148475074256982964816314,
%U A300593 263023328027145941803648512,590040725672004981627313856146,1648073412972421008768279297745708,5648002661974709728272920853918580200,23444503972399728196572891896057248430080
%N A300593 O.g.f. A(x) satisfies: [x^n] exp( n^2 * A(x) ) = n^3 * [x^(n-1)] exp( n^2 * A(x) ) for n>=1.
%C A300593 Compare to: [x^n] exp( n^2 * x ) = n * [x^(n-1)] exp( n^2 * x ) for n>=1.
%C A300593 It is conjectured that this sequence consists entirely of integers.
%H A300593 Paul D. Hanna, <a href="/A300593/b300593.txt">Table of n, a(n) for n = 1..200</a>
%F A300593 O.g.f. equals the logarithm of the e.g.f. of A300592.
%F A300593 a(n) ~ c * n!^3, where c = 3.10566781078993955626127892108166... - _Vaclav Kotesovec_, Oct 14 2020
%e A300593 O.g.f.: A(x) = x + 6*x^2 + 216*x^3 + 18016*x^4 + 2718575*x^5 + 667151244*x^6 + 249904389518*x^7 + 136335045655680*x^8 + 104258627494173747*x^9 ...
%e A300593 where
%e A300593 exp(A(x)) = 1 + x + 13*x^2/2! + 1333*x^3/3! + 438073*x^4/4! + 328561681*x^5/5! + 482408372341*x^6/6! + 1262989939509733*x^7/7! + ... + A300592(n)*x^n/n! + ...
%e A300593 such that: [x^n] exp( n^2 * A(x) ) = n^3 * [x^(n-1)] exp( n^2 * A(x) ).
%o A300593 (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)^3*V[#A-1] - V[#A])/(#A-1)^2 ); polcoeff( log(Ser(A)), n)}
%o A300593 for(n=1, 30, print1(a(n), ", "))
%Y A300593 Cf. A300592, A296171, A300591, A300595.
%K A300593 nonn
%O A300593 1,2
%A A300593 _Paul D. Hanna_, Mar 09 2018