A224102 - OEIS

%I #37 Feb 26 2023 18:11:52

%S 1,-1,13,-43,5647,-3401,2763977,-10326059,876576493,-1665984623,

%T 1156096889861,-2220482068331,75970695882225719,-1088498788093641,

%U 855021689397409453,-3324381371618385007,4010325276269988793421

%N Numerators of poly-Cauchy numbers of the second kind hat c_n^(2).

%C 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).

%H Vincenzo Librandi, <a href="/A224102/b224102.txt">Table of n, a(n) for n = 0..300</a>

%H Takao Komatsu, <a href="http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1806-06.pdf">Poly-Cauchy numbers</a>, RIMS Kokyuroku 1806 (2012)

%H Takao Komatsu, <a href="http://link.springer.com/article/10.1007/s11139-012-9452-0">Poly-Cauchy numbers with a q parameter</a>, Ramanujan J. 31 (2013), 353-371.

%H Takao Komatsu, <a href="http://doi.org/10.2206/kyushujm.67.143">Poly-Cauchy numbers</a>, Kyushu J. Math. 67 (2013), 143-153.

%H T. Komatsu, V. Laohakosol, K. Liptai, <a href="http://dx.doi.org/10.1155/2013/179841">A generalization of poly-Cauchy numbers and its properties</a>, Abstract and Applied Analysis, Volume 2013, Article ID 179841, 8 pages.

%H Takao Komatsu, FZ Zhao, <a href="http://arxiv.org/abs/1603.06725">The log-convexity of the poly-Cauchy numbers</a>, arXiv preprint arXiv:1603.06725, 2016

%t Table[Numerator[Sum[StirlingS1[n, k] (-1)^k/ (k + 1)^2, {k, 0, n}]], {n, 0,

%t   25}]

%o (PARI) a(n) = numerator(sum(k=0, n, stirling(n, k, 1)*(-1)^k/(k+1)^2)); \\ _Michel Marcus_, Nov 14 2015

%Y Cf. A002657, A223899, A219247 (denominators).

%K sign,frac

%O 0,3

%A _Takao Komatsu_, Mar 31 2013