A328286 Expansion of e.g.f. -log(1 - x - x^2/2).

1, 2, 5, 21, 114, 780, 6390, 61110, 667800, 8210160, 112152600, 1685237400, 27624920400, 490572482400, 9381882510000, 192238348302000, 4201639474032000, 97572286427616000, 2399151995223984000, 62268748888378032000, 1701213856860117600000

Offset: 1

Author

Ilya Gutkovskiy, Oct 11 2019

Keywords

• nonn

Crossrefs

Cf. A009014, A039647, A080040, A080599 (exponential transform).

Programs

Maple

b:= proc(n) b(n):= n! * (<<1|1>, <1/2|0>>^n)[1, 1] 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=1..25);  # Alois P. Heinz, Oct 11 2019

Mathematica

nmax = 21; CoefficientList[Series[-Log[1 - x - x^2/2], {x, 0, nmax}], x] Range[0, nmax]! // Rest
FullSimplify[Table[(n - 1)! ((1 - Sqrt[3])^n + (1 + Sqrt[3])^n)/2^n, {n, 1, 21}]]

PARI

my(x='x+O('x^25)); Vec(serlaplace(-log(1 - x - x^2/2))) \\ Michel Marcus, Oct 11 2019

Formula

a(n) = (n - 1)! * ((1 - sqrt(3))^n + (1 + sqrt(3))^n) / 2^n.

D-finite with recurrence +2*a(n) +2*(-n+1)*a(n-1) -(n-1)*(n-2)*a(n-2)=0. - R. J. Mathar, Aug 20 2021