A224097 - OEIS

%I #40 Feb 26 2023 13:20:27

%S 1,1,-19,89,-46261,23323,-114895757,760567603,-174446569403,

%T 302339104957,-2125170096355349,3788248001789087,

%U -1573899862241140688567,317684785943639774839,-2242333884754953400123

%N Numerators of poly-Cauchy numbers c_n^(3).

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

%H Vincenzo Librandi, <a href="/A224097/b224097.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]/ (k + 1)^3, {k, 0, n}]], {n, 0, 25}]

%o (PARI) a(n) = numerator(sum(k=0, n,stirling(n, k, 1)/(k+1)^3)); \\ _Michel Marcus_, Nov 15 2015

%Y Cf. A006232, A222636, A224095, A224096 (denominators).

%K sign,frac

%O 0,3

%A _Takao Komatsu_, Mar 31 2013