cp's OEIS Frontend

A304314 Logarithmic derivative of F(x) that satisfies: [x^n] exp( n^4 * x ) / F(x) = 0 for n>0.

%I A304314 #19 Aug 31 2020 10:57:21
%S A304314 1,225,229000,612243125,3367384031526,33056423981177346,
%T A304314 527146092112494861420,12764850938355048224394925,
%U A304314 446065249480005516657138106375,21615893741029073481369412949207860,1406758471936562034421316174257309550136,119755662436589797897149020637183902177930534
%N A304314 Logarithmic derivative of F(x) that satisfies: [x^n] exp( n^4 * x ) / F(x) = 0 for n>0.
%C A304314 Conjecture: a(n) is the number of connected n-state finite automata with 4 inputs.
%C A304314 Equals row 4 of table A304321.
%H A304314 Paul D. Hanna, <a href="/A304314/b304314.txt">Table of n, a(n) for n = 0..200</a>
%F A304314 Logarithmic derivative of the o.g.f. of A304324.
%F A304314 For n>=1, a(n) = B_{n+1}((n+1)^4-0!*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 A304314 a(n) ~ sqrt(1-c) * 4^(4*(n+1)) * n^(3*n + 7/2) / (sqrt(2*Pi) * c^(n+1) * (4-c)^(3*(n+1)) * exp(3*n)), where c = -LambertW(-4*exp(-4)). - _Vaclav Kotesovec_, Aug 31 2020
%e A304314 O.g.f.: L(x) = 1 + 225*x + 229000*x^2 + 612243125*x^3 + 3367384031526*x^4 + 33056423981177346*x^5 + 527146092112494861420*x^6 + ...
%e A304314 such that L(x) = F'(x)/F(x) where F(x) is the o.g.f. of A304324 :
%e A304314 F(x) = 1 + x + 113*x^2 + 76446*x^3 + 153143499*x^4 + 673638499100*x^5 + 5510097691767062*x^6 + 75312181798660695788*x^7 + ... + A304324(n)*x^n + ...
%e A304314 which satisfies [x^n] exp( n^4 * x ) / F(x) = 0 for n>0.
%t A304314 m = 25;
%t A304314 F = 1 + Sum[c[k] x^k, {k, m}];
%t A304314 s[n_] := Solve[SeriesCoefficient[Exp[n^4*x]/F, {x, 0, n}] == 0][[1]];
%t A304314 Do[F = F /. s[n], {n, m}];
%t A304314 CoefficientList[D[F, x]/F + O[x]^m, x] (* _Jean-François Alcover_, May 21 2018 *)
%o A304314 (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 A304314 for(n=0,25, print1( a(n),", "))
%Y A304314 Cf. A304324, A304321, A304312, A304313, A304315.
%K A304314 nonn
%O A304314 0,2
%A A304314 _Paul D. Hanna_, May 11 2018