A300488 a(n) = n! * [x^n] -exp(n*x)*log(1 - x)/(1 - x).

0, 1, 7, 65, 770, 11149, 191124, 3788469, 85281552, 2149582761, 59983774240, 1835925702137, 61157508893568, 2202760340194517, 85303050939131648, 3534478528925155725, 156026612737389987840, 7310587974761946511761, 362356607517279564386304, 18943214212273585171456753

Offset: 0

Author

Ilya Gutkovskiy, Mar 07 2018

Keywords

• nonn

Examples

			The table of coefficients of x^k in expansion of e.g.f. -exp(n*x)*log(1 - x)/(1 - x) begins:
n = 0: (0), 1,   3,   11,    50,     274,  ...
n = 1:  0, (1),  5,   23,   116,     669,  ...
n = 2:  0,  1,  (7),  41,   242,    1534,  ...
n = 3:  0,  1,   9,  (65),  452,    3229,  ...
n = 4:  0,  1,  11,   95,  (770),   6234,  ...
n = 5:  0,  1,  13,  131,  1220,  (11149), ...
...
This sequence is the main diagonal of the table.
		

Links

• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000254, A065456, A073596.

Programs

Mathematica

Table[n! SeriesCoefficient[-Exp[n x] Log[1 - x]/(1 - x), {x, 0, n}], {n, 0, 19}]
Table[Sum[n^(n - k) Binomial[n,k] k! HarmonicNumber[k], {k, 1, n}], {n, 0, 19}]

Formula

a(n) = Sum_{k=1..n} n^(n-k)*binomial(n,k)*k!*H(k), where H(k) is the k-th harmonic number.