A224104 Numerators of poly-Cauchy numbers of the second kind hat c_n^(3).

1, -1, 35, -217, 135989, -236881, 435876493, -3174551347, 790667708347, -1473406853309, 11050163107919893, -20886680047664287, 9154917271574968829623, -277315386220087376401, 803143323197313772705

Offset: 0

Author

Takao Komatsu, Mar 31 2013

Keywords

• sign
• frac

Comments

The poly-Cauchy numbers of the second kind hat c_n^(k) can be expressed in terms of the (unsigned) Stirling numbers of the first kind: hat c_n^(k) = (-1)^n*sum(abs(stirling1(n,m))/(m+1)^k, m=0..n).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Takao Komatsu, Poly-Cauchy numbers, RIMS Kokyuroku 1806 (2012)
• Takao Komatsu, Poly-Cauchy numbers with a q parameter, Ramanujan J. 31 (2013), 353-371.
• Takao Komatsu, Poly-Cauchy numbers, Kyushu J. Math. 67 (2013), 143-153.
• T. Komatsu, V. Laohakosol, K. Liptai, A generalization of poly-Cauchy numbers and its properties, Abstract and Applied Analysis, Volume 2013, Article ID 179841, 8 pages.
• Takao Komatsu, FZ Zhao, The log-convexity of the poly-Cauchy numbers, arXiv preprint arXiv:1603.06725, 2016

Crossrefs

Cf. A002657, A223901, A224102, A224103 (denominators).

Programs

Mathematica

Table[Numerator[Sum[StirlingS1[n, k] (-1)^k/ (k + 1)^3, {k, 0, n}]], {n, 0,
  25}]

PARI

a(n) = numerator(sum(k=0, n, stirling(n, k, 1)*(-1)^k/(k+1)^3)); \\ Michel Marcus, Nov 14 2015