A304316 Logarithmic derivative of F(x) that satisfies: [x^n] exp( n*(n-1) * x ) / F(x) = 0 for n>0.

4, 72, 1736, 53040, 1961728, 85062432, 4225904800, 236455369344, 14705880874944, 1005982098054912, 75048224139686912, 6062679436944758784, 527187725605767366144, 49092882744958427976192, 4874131922792403196021248, 513942386047796079510884352, 57356122407632751143615036416, 6754087907265415509369502427136, 836924235604443592471459956156416

Offset

1,1

Links

Paul D. Hanna, Table of n, a(n) for n = 1..400

Formula

Logarithmic derivative of the o.g.f. of A304318.

a(n) ~ sqrt(1-c) * 2^(2*n + 1) * n^(n + 3/2) / (sqrt(Pi) * c^(n + 1/2) * (2-c)^(n+1) * exp(n)), where c = -A226775 = -LambertW(-2*exp(-2)). - Vaclav Kotesovec, Aug 31 2020

Example

O.g.f.: L(x) = 4*x + 72*x^2 + 1736*x^3 + 53040*x^4 + 1961728*x^5 + 85062432*x^6 + 4225904800*x^7 + 236455369344*x^8 + 14705880874944*x^9 + 1005982098054912*x^10 + ...
such that L(x) = F'(x)/F(x) where F(x) is the o.g.f. of A304324 :
F(x) = 1 + 2*x^2 + 24*x^3 + 436*x^4 + 10656*x^5 + 328112*x^6 + 12183456*x^7 + 529242224*x^8 + 26309617536*x^9 + 1472135847072*x^10 + ... + A304318(n)*x^n + ...
which satisfies [x^n] exp( n*(n-1) * 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*(n - 1)*x]/F, {x, 0, n}] == 0][[1]];
Do[F = F /. s[n], {n, m}];
CoefficientList[D[F, x]/F + O[x]^m, x] // Rest (* 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)*(m-2) +x^2*O(x^m)) / Ser(A) )[m] ); L = Vec(Ser(A)'/Ser(A)); L[n]}
for(n=1, 25, print1( a(n), ", "))

Crossrefs

Cf. A304318, A304317, A304321.

Sequence in context: A201976 A328426 A176901 * A377356 A186415 A211038

Adjacent sequences: A304313 A304314 A304315 * A304317 A304318 A304319

Keyword

nonn

Author

Paul D. Hanna, May 11 2018

Status

approved