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A304317 Logarithmic derivative of F(x) that satisfies: [x^n] exp( n*(n+1) * x ) / F(x) = 0 for n>0.

%I A304317 #12 Aug 31 2020 06:30:50
%S A304317 2,16,260,6200,191832,7235152,320372320,16243028896,926219213216,
%T A304317 58608051937536,4072302306624576,308103163039452416,
%U A304317 25206121081157341184,2216835666085110716928,208552041718766935142400,20896682922005650384144896,2221700463749100463405564416,249800738062720558095843241984,29615243677328447562465854639104
%N A304317 Logarithmic derivative of F(x) that satisfies: [x^n] exp( n*(n+1) * x ) / F(x) = 0 for n>0.
%H A304317 Paul D. Hanna, <a href="/A304317/b304317.txt">Table of n, a(n) for n = 0..400</a>
%F A304317 Logarithmic derivative of the o.g.f. of A304319.
%F A304317 a(n) ~ sqrt(1-c) * 2^(2*n + 2) * n^(n + 3/2) / (sqrt(Pi) * c^(n + 3/2) * (2-c)^(n+1) * exp(n)), where c = -A226775 = -LambertW(-2*exp(-2)). - _Vaclav Kotesovec_, Aug 31 2020
%e A304317 O.g.f.: L(x) = 2 + 16*x + 260*x^2 + 6200*x^3 + 191832*x^4 + 7235152*x^5 + 320372320*x^6 + 16243028896*x^7 + 926219213216*x^8 + 58608051937536*x^9 + 4072302306624576*x^10 + ...
%e A304317 such that L(x) = F'(x)/F(x) where F(x) is the o.g.f. of A304319:
%e A304317 F(x) = 1 + 2*x + 10*x^2 + 104*x^3 + 1772*x^4 + 42408*x^5 + 1303504*x^6 + 48736000*x^7 + 2139552016*x^8 + 107629121888*x^9 + 6094743943584*x^10 + ... + A304319(n)*x^n + ...
%e A304317 which satisfies [x^n] exp( n*(n+1) * x ) / F(x) = 0 for n>0.
%t A304317 m = 25;
%t A304317 F = 1 + Sum[c[k] x^k, {k, m}];
%t A304317 s[n_] := Solve[SeriesCoefficient[Exp[n*(n + 1)*x]/F, {x, 0, n}] == 0][[1]];
%t A304317 Do[F = F /. s[n], {n, m}];
%t A304317 CoefficientList[D[F, x]/F + O[x]^m, x] (* _Jean-François Alcover_, May 21 2018 *)
%o A304317 (PARI) {a(n) = my(A=[1],L); for(i=0, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*m*(m-1) +x^2*O(x^m)) / Ser(A) )[m] ); L = Vec(Ser(A)'/Ser(A)); L[n+1]}
%o A304317 for(n=0,25, print1( a(n),", "))
%Y A304317 Cf. A304319, A304316, A304321.
%K A304317 nonn
%O A304317 0,1
%A A304317 _Paul D. Hanna_, May 11 2018