A304317 Logarithmic derivative of F(x) that satisfies: [x^n] exp( n*(n+1) * x ) / F(x) = 0 for n>0.

2, 16, 260, 6200, 191832, 7235152, 320372320, 16243028896, 926219213216, 58608051937536, 4072302306624576, 308103163039452416, 25206121081157341184, 2216835666085110716928, 208552041718766935142400, 20896682922005650384144896, 2221700463749100463405564416, 249800738062720558095843241984, 29615243677328447562465854639104

Offset

0,1

Links

Paul D. Hanna, Table of n, a(n) for n = 0..400

Formula

Logarithmic derivative of the o.g.f. of A304319.

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

Example

O.g.f.: L(x) = 2 + 16*x + 260*x^2 + 6200*x^3 + 191832*x^4 + 7235152*x^5 + 320372320*x^6 + 16243028896*x^7 + 926219213216*x^8 + 58608051937536*x^9 + 4072302306624576*x^10 + ...
such that L(x) = F'(x)/F(x) where F(x) is the o.g.f. of A304319:
F(x) = 1 + 2*x + 10*x^2 + 104*x^3 + 1772*x^4 + 42408*x^5 + 1303504*x^6 + 48736000*x^7 + 2139552016*x^8 + 107629121888*x^9 + 6094743943584*x^10 + ... + A304319(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] (* 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*(m-1) +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. A304319, A304316, A304321.

Sequence in context: A090305 A358145 A246739 * A351918 A326272 A283685

Adjacent sequences: A304314 A304315 A304316 * A304318 A304319 A304320

Keyword

nonn

Author

Paul D. Hanna, May 11 2018

Status

approved