A380338 Expansion of e.g.f. log(1 - x * log(1 - x)).

0, 0, 2, 3, -4, -30, 54, 1260, 3856, -36288, -279000, 2970000, 56725008, 109343520, -5495740992, -26086263840, 1293641890560, 21771049466880, -45508965806592, -4589738336217600, 10493846174810880, 2423866077943511040, 34328754265480012800, -358930542362135546880

Offset: 0

Author

Seiichi Manyama, Jan 21 2025

Keywords

• sign

Crossrefs

Cf. A052804, A052858.

Cf. A089064, A380339.

Cf. A001048, A375684.

Programs

PARI

my(N=30, x='x+O('x^N)); concat([0, 0], Vec(serlaplace(log(1-x*log(1-x)))))

PARI

a(n) = n!*sum(k=1, n\2, (-1)^(k-1)*(k-1)!*abs(stirling(n-k, k, 1))/(n-k)!);

Formula

a(n) = n! * Sum_{k=1..floor(n/2)} (-1)^(k-1) * (k-1)! * |Stirling1(n-k,k)|/(n-k)!.

a(0) = a(1) = 0; a(n) = n * (n-2)! - Sum_{k=2..n-1} k * (k-2)! * binomial(n-1,k) * a(n-k).