A306325 Expansion of e.g.f. log(1 + exp(x)*x*(1 + 7*x + 6*x^2 + x^3)).

0, 1, 15, 35, -650, -5251, 83376, 1623439, -19261584, -836109351, 5365104400, 636771444011, 561938325312, -661384866976523, -7128491581221360, 879709224738485415, 21742632225425026816, -1413667730904479933647, -64871991410092201623024, 2556051301724027073500035, 212244727356899863738042560

Offset: 0

Author

Ilya Gutkovskiy, Feb 07 2019

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..384

Crossrefs

Cf. A000583, A033464, A279637, A300452.

Programs

Maple

a:=series(log(1 + exp(x)*x*(1 + 7*x + 6*x^2 + x^3)),x=0,21): seq(n!*coeff(a, x, n),n=0..20); # Paolo P. Lava, Mar 26 2019

Mathematica

nmax = 20; CoefficientList[Series[Log[1 + Exp[x] x (1 + 7 x + 6 x^2 + x^3)], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = n^4 - Sum[Binomial[n, k] (n - k)^4 k a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 20}]

Formula

E.g.f.: log(1 + Sum_{k>=1} k^4*x^k/k!).

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