A304313 Logarithmic derivative of F(x) that satisfies: [x^n] exp( n^3 * x ) / F(x) = 0 for n>0.

1, 49, 6877, 1854545, 807478656, 514798204147, 451182323794896, 519961864703259753, 762210147961330421167, 1384945048774500147047194, 3055115321627096660341307614, 8043516699726480852467167758419, 24915939138210507189761922944830006, 89709850983809128394441772076036629240, 371523831948166269091257380175120352465872

Offset

0,2

Comments

Is this sequence essentially the same as A006692?

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

Equals row 3 of table A304321.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

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

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

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

Example

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 + ...
such that L(x) = F'(x)/F(x) where F(x) is the o.g.f. of A304322 :
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 + ...
which satisfies [x^n] exp( n^3 * x ) / F(x) = 0 for n>0.

Mathematica

m = 25;
F = 1 + Sum[c[k] x^k, {k, m}];
s[n_] := Solve[SeriesCoefficient[Exp[n^3*x]/F, {x, 0, n}] == 0][[1]];
Do[F = F /. s[n], {n, m}];
CoefficientList[D[F, x]/F + O[x]^m, x] (* Jean-François Alcover, May 21 2018 *)

Prog

(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]}
for(n=0, 25, print1( a(n), ", "))

Crossrefs

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

Sequence in context: A127861 A130416 A006692 * A283788 A206247 A180273

Adjacent sequences: A304310 A304311 A304312 * A304314 A304315 A304316

Keyword

nonn

Author

Paul D. Hanna, May 11 2018

Status

approved