A304313 - OEIS

%I #16 Aug 31 2020 10:57:02

%S 1,49,6877,1854545,807478656,514798204147,451182323794896,

%T 519961864703259753,762210147961330421167,1384945048774500147047194,

%U 3055115321627096660341307614,8043516699726480852467167758419,24915939138210507189761922944830006,89709850983809128394441772076036629240,371523831948166269091257380175120352465872

%N Logarithmic derivative of F(x) that satisfies: [x^n] exp( n^3 * x ) / F(x) = 0 for n>0.

%C Is this sequence essentially the same as A006692?

%C Conjecture: a(n) is the number of connected n-state finite automata with 3 inputs (A006692).

%C Equals row 3 of table A304321.

%H Paul D. Hanna, <a href="/A304313/b304313.txt">Table of n, a(n) for n = 0..300</a>

%F Logarithmic derivative of the o.g.f. of A304323.

%F For n>=1, a(n) = B_{n+1}((n+1)^3-0!*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 a(n) ~ sqrt(1-c) * 3^(3*(n+1)) * n^(2*n + 5/2) / (sqrt(2*Pi) * c^(n+1) * (3-c)^(2*(n+1)) * exp(2*n)), where c = -LambertW(-3*exp(-3)). - _Vaclav Kotesovec_, Aug 31 2020

%e O.g.f.: L(x) = 1 + 49*x + 6877*x^2 + 1854545*x^3 + 807478656*x^4 + 514798204147*x^5 + 451182323794896*x^6 + 519961864703259753*x^7 + ...

%e such that L(x) = F'(x)/F(x) where F(x) is the o.g.f. of A304322 :

%e F(x) = 1 + x + 25*x^2 + 2317*x^3 + 466241*x^4 + 162016980*x^5 + 85975473871*x^6 + 64545532370208*x^7 + 65062315637060121*x^8 + ... + A304323(n)*x^n + ...

%e which satisfies [x^n] exp( n^3 * x ) / F(x) = 0 for n>0.

%t m = 25;

%t F = 1 + Sum[c[k] x^k, {k, m}];

%t s[n_] := Solve[SeriesCoefficient[Exp[n^3*x]/F, {x, 0, n}] == 0][[1]];

%t Do[F = F /. s[n], {n, m}];

%t CoefficientList[D[F, x]/F + O[x]^m, x] (* _Jean-François Alcover_, May 21 2018 *)

%o (PARI) {a(n) = my(A=[1],L); for(i=0, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)^3 +x^2*O(x^m)) / Ser(A) )[m] ); L = Vec(Ser(A)'/Ser(A)); L[n+1]}

%o for(n=0,25, print1( a(n),", "))

%Y Cf. A304323, A006692, A304321, A304312, A304314, A304315.

%K nonn

%O 0,2

%A _Paul D. Hanna_, May 11 2018