A354686 a(n) = n! * Sum_{k=1..n} Stirling1(n,k) * H(k), where H(k) is the k-th harmonic number.

0, 1, 1, -4, 38, -646, 17124, -651120, 33563760, -2251415376, 190506294720, -19843054116480, 2494435702953600, -372324067662349440, 65089674982557308160, -13172994619821785548800, 3055455516855073351219200, -805168341051328705189939200

Offset: 0

Author

Ilya Gutkovskiy, Jun 03 2022

Keywords

• sign

Crossrefs

Cf. A001008, A002805, A087751, A222059, A302547, A354685.

Programs

Mathematica

Table[n! Sum[StirlingS1[n, k] HarmonicNumber[k], {k, 1, n}], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Sum[HarmonicNumber[k] Log[1 + x]^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / n!^2 = Sum_{n>=1} H(n) * log(1+x)^n / n!.