A328054 Expansion of e.g.f. log(1 + x / (1 - x)^2).

0, 1, 3, 8, 18, 24, 0, 720, 15120, 161280, 1088640, 3628800, 0, 479001600, 18681062400, 348713164800, 3923023104000, 20922789888000, 0, 6402373705728000, 364935301226496000, 9731608032706560000, 153272826515128320000, 1124000727777607680000, 0, 620448401733239439360000

Offset: 0

Author

Ilya Gutkovskiy, Oct 03 2019

Keywords

• nonn

Comments

Logarithmic transform of A001563.

Links

• Robert Israel, Table of n, a(n) for n = 0..450

Crossrefs

Cf. A001563, A008588 (positions of 0's), A009306, A082579, A328055.

Programs

Maple

b:= proc(n) option remember; n*n! end:
a:= proc(n) option remember; `if`(n=0, 0, b(n)-
      add(binomial(n, j)*j*b(n-j)*a(j), j=1..n-1)/n)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 04 2019

Mathematica

nmax = 25; CoefficientList[Series[Log[1 + x/(1 - x)^2], {x, 0, nmax}], x] Range[0, nmax]!
Join[{0}, Table[2 (n - 1)! (1 - Cos[Pi n/3]), {n, 1, 25}]]

PARI

my(x='x+O('x^30)); concat(0, Vec(serlaplace(log(1 + x / (1 - x)^2)))) \\ Michel Marcus, Oct 04 2019

Formula

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

D-finite with recurrence a(n+3) = n*(n+1)*(n+2)*a(n) - 2*(n+2)*(n+1)*a(n+1) + 2*(n+2)*a(n+2). - Robert Israel, Jan 16 2023