A224106 Numerators of poly-Cauchy numbers of the second kind hat c_n^(4).

1, -1, 97, -1147, 3472243, -653983, 74118189437, -1058923294571, 777910456216513, -285577840060819, 23240203016832136201, -216925341603548096639, 1222007019804929270080450811

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, and K. Liptai, A generalization of poly-Cauchy numbers and its properties, Abstract and Applied Analysis, Volume 2013, Article ID 179841, 8 pages.
• Takao Komatsu and F.-Z. Zhao, The log-convexity of the poly-Cauchy numbers, arXiv preprint arXiv:1603.06725 [math.NT], 2016.

Crossrefs

Cf. A002657, A223902, A224105, A114102, A224104, A224105 (denominators).

Programs

Mathematica

Table[Numerator[Sum[StirlingS1[n, k] (-1)^k/ (k + 1)^4, {k, 0, n}]], {n, 0,
  25}]

PARI

a(n) = numerator(sum(k=0, n,(-1)^k*stirling(n, k, 1)/(k+1)^4)); \\ Michel Marcus, Nov 15 2015