A384522 Expansion of e.g.f. 4/(7 - 3*exp(4*x)).

1, 3, 30, 426, 8040, 189768, 5375280, 177632976, 6708685440, 285038686848, 13456362881280, 698786099602176, 39586707755811840, 2429498408440009728, 160571526535426529280, 11370607719608891467776, 858870213271187908362240, 68928740686010010238353408

Offset

0,2

Links

Table of n, a(n) for n=0..17.

Wikipedia, Polylogarithm.

Formula

a(n) = (-4)^(n+1)/7 * Li_{-n}(7/3), where Li_{n}(x) is the polylogarithm function.

a(n) = 4^(n+1)/7 * Sum_{k>=0} k^n * (3/7)^k.

a(n) = Sum_{k=0..n} 3^k * 4^(n-k) * k! * Stirling2(n,k).

a(n) = (3/7) * Sum_{k=0..n} 7^k * (-4)^(n-k) * k! * Stirling2(n,k) for n > 0.

a(0) = 1; a(n) = 3 * Sum_{k=1..n} 4^(k-1) * binomial(n,k) * a(n-k).

a(0) = 1; a(n) = 3 * a(n-1) + 7 * Sum_{k=1..n-1} (-4)^(k-1) * binomial(n-1,k) * a(n-k).

Mathematica

With[{nn=20}, CoefficientList[Series[4/(7-3Exp[4x]), {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Jun 21 2025 *)

Prog

(PARI) a(n) = (-4)^(n+1)*polylog(-n, 7/3)/7;

Crossrefs

Cf. A094419, A384514, A384521, A384523, A384524.

Sequence in context: A294240 A392837 A007004 * A354291 A354287 A276361

Adjacent sequences: A384519 A384520 A384521 * A384523 A384524 A384525

Keyword

nonn

Author

Seiichi Manyama, Jun 01 2025

Status

approved