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A304316 Logarithmic derivative of F(x) that satisfies: [x^n] exp( n*(n-1) * x ) / F(x) = 0 for n>0.

%I A304316 #13 Aug 31 2020 07:01:40
%S A304316 4,72,1736,53040,1961728,85062432,4225904800,236455369344,
%T A304316 14705880874944,1005982098054912,75048224139686912,
%U A304316 6062679436944758784,527187725605767366144,49092882744958427976192,4874131922792403196021248,513942386047796079510884352,57356122407632751143615036416,6754087907265415509369502427136,836924235604443592471459956156416
%N A304316 Logarithmic derivative of F(x) that satisfies: [x^n] exp( n*(n-1) * x ) / F(x) = 0 for n>0.
%H A304316 Paul D. Hanna, <a href="/A304316/b304316.txt">Table of n, a(n) for n = 1..400</a>
%F A304316 Logarithmic derivative of the o.g.f. of A304318.
%F A304316 a(n) ~ sqrt(1-c) * 2^(2*n + 1) * n^(n + 3/2) / (sqrt(Pi) * c^(n + 1/2) * (2-c)^(n+1) * exp(n)), where c = -A226775 = -LambertW(-2*exp(-2)). - _Vaclav Kotesovec_, Aug 31 2020
%e A304316 O.g.f.: L(x) = 4*x + 72*x^2 + 1736*x^3 + 53040*x^4 + 1961728*x^5 + 85062432*x^6 + 4225904800*x^7 + 236455369344*x^8 + 14705880874944*x^9 + 1005982098054912*x^10 + ...
%e A304316 such that L(x) = F'(x)/F(x) where F(x) is the o.g.f. of A304324 :
%e A304316 F(x) = 1 + 2*x^2 + 24*x^3 + 436*x^4 + 10656*x^5 + 328112*x^6 + 12183456*x^7 + 529242224*x^8 + 26309617536*x^9 + 1472135847072*x^10 + ... + A304318(n)*x^n + ...
%e A304316 which satisfies [x^n] exp( n*(n-1) * x ) / F(x) = 0 for n>0.
%t A304316 m = 25;
%t A304316 F = 1 + Sum[c[k] x^k, {k, m}];
%t A304316 s[n_] := Solve[SeriesCoefficient[Exp[n*(n - 1)*x]/F, {x, 0, n}] == 0][[1]];
%t A304316 Do[F = F /. s[n], {n, m}];
%t A304316 CoefficientList[D[F, x]/F + O[x]^m, x] // Rest (* _Jean-François Alcover_, May 21 2018 *)
%o A304316 (PARI) {a(n) = my(A=[1],L); for(i=0, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)*(m-2) +x^2*O(x^m)) / Ser(A) )[m] ); L = Vec(Ser(A)'/Ser(A)); L[n]}
%o A304316 for(n=1,25, print1( a(n),", "))
%Y A304316 Cf. A304318, A304317, A304321.
%K A304316 nonn
%O A304316 1,1
%A A304316 _Paul D. Hanna_, May 11 2018