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A304315 Logarithmic derivative of F(x) that satisfies: [x^n] exp( n^5 * x ) / F(x) = 0 for n>0.

%I A304315 #17 Aug 31 2020 10:57:42
%S A304315 1,961,6737401,172342090401,11657788116175751,1722786509653595220757,
%T A304315 489506033977061086758261063,243968979437942649897623460813009,
%U A304315 199025593654123221838381793032781035510,251774439716905627952289102887999425054599511,472942802381336010263584088374665504251010554412128,1273071332950625956697135571575613091625334028239417955701
%N A304315 Logarithmic derivative of F(x) that satisfies: [x^n] exp( n^5 * x ) / F(x) = 0 for n>0.
%C A304315 Conjecture: a(n) is the number of connected n-state finite automata with 5 inputs.
%C A304315 Equals row 5 of table A304321.
%H A304315 Paul D. Hanna, <a href="/A304315/b304315.txt">Table of n, a(n) for n = 0..200</a>
%F A304315 Logarithmic derivative of the o.g.f. of A304325.
%F A304315 For n>=1, a(n) = B_{n+1}((n+1)^5-0!*a(0),-1!*a(1),...,-(n-1)!*a(n-1),0) / n!, where B_{n+1}(...) is the (n+1)-st complete exponential Bell polynomial. - _Max Alekseyev_, Jun 18 2018
%F A304315 a(n) ~ sqrt(1-c) * 5^(5*(n+1)) * n^(4*n + 9/2) / (sqrt(2*Pi) * c^(n+1) * (5-c)^(4*(n+1)) * exp(4*n)), where c = -LambertW(-5*exp(-5)). - _Vaclav Kotesovec_, Aug 31 2020
%e A304315 O.g.f.: L(x) = 1 + 961*x + 6737401*x^2 + 172342090401*x^3 + 11657788116175751*x^4 + 1722786509653595220757*x^5 + 489506033977061086758261063*x^6 + ...
%e A304315 such that L(x) = F'(x)/F(x) where F(x) is the o.g.f. of A304325 :
%e A304315 F(x) = 1 + x + 481*x^2 + 2246281*x^3 + 43087884081*x^4 + 2331601789103231*x^5 + 287133439746933073357*x^6 + 69929721774643572422651223*x^7 + ... + A304325(n)*x^n + ...
%e A304315 which satisfies [x^n] exp( n^5 * x ) / F(x) = 0 for n>0.
%t A304315 m = 25;
%t A304315 F = 1 + Sum[c[k] x^k, {k, m}];
%t A304315 s[n_] := Solve[SeriesCoefficient[Exp[n^5*x]/F, {x, 0, n}] == 0][[1]];
%t A304315 Do[F = F /. s[n], {n, m}];
%t A304315 CoefficientList[D[F, x]/F + O[x]^m, x] (* _Jean-François Alcover_, May 21 2018 *)
%o A304315 (PARI) {a(n) = my(A=[1],L); for(i=0, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)^4 +x^2*O(x^m)) / Ser(A) )[m] ); L = Vec(Ser(A)'/Ser(A)); L[n+1]}
%o A304315 for(n=0,25, print1( a(n),", "))
%Y A304315 Cf. A304325, A304321, A304312, A304313, A304314.
%K A304315 nonn
%O A304315 0,2
%A A304315 _Paul D. Hanna_, May 11 2018