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A222636 Poly-Cauchy numbers c_n^(-3).

%I A222636 #60 Apr 16 2025 09:02:31
%S A222636 1,8,19,-1,-10,48,-234,1302,-8328,60672,-497688,4547448,-45846864,
%T A222636 505862064,-6065584128,78555965184,-1093053332736,16264215348480,
%U A222636 -257730606190080,4333624828853760,-77067187081620480,1445257352902763520,-28505367984508416000
%N A222636 Poly-Cauchy numbers c_n^(-3).
%C A222636 Definition of poly-Cauchy numbers in A222627.
%H A222636 Vincenzo Librandi, <a href="/A222636/b222636.txt">Table of n, a(n) for n = 0..300</a>
%H A222636 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 A222636 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 A222636 Takao Komatsu, <a href="http://doi.org/10.2206/kyushujm.67.143">Poly-Cauchy numbers</a>, Kyushu J. Math. 67 (2013), 143-153.
%H A222636 Takao Komatsu, <a href="https://doi.org/10.22436/jnsa.012.12.05">Some recurrence relations of poly-Cauchy numbers</a>, J. Nonlinear Sci. Appl., (2019) Vol. 12, Issue 12, 829-845.
%H A222636 M. Z. Spivey,<a href="http://dx.doi.org/10.1016/j.disc.2007.03.052">Combinatorial sums and finite differences</a>, Discr. Math. 307 (24) (2007) 3130-3146.
%H A222636 Wikipedia, <a href="http://en.wikipedia.org/wiki/Stirling_transform">Stirling transform</a>
%F A222636 a(n) = Sum_{k=0..n} Stirling1(n,k) * (k+1)^3.
%F A222636 E.g.f.: (1 + x) * (1 + 7 * log(1 + x) + 6 * log(1 + x)^2 + log(1 + x)^3). - _Ilya Gutkovskiy_, Aug 10 2021
%F A222636 E.g.f.: Sum_{k>=0} (k+1)^3 * log(1+x)^k / k!. - _Seiichi Manyama_, Apr 14 2025
%t A222636 Table[Sum[StirlingS1[n, k] (k + 1)^3, {k, 0, n}], {n, 0, 25}]
%o A222636 (Magma) [&+[StirlingFirst(n,k)*(k+1)^3: k in [0..n]]: n in [0..25]]; // _Bruno Berselli_, Mar 28 2013
%o A222636 (PARI) a(n) = sum(k=0, n, stirling(n, k, 1)*(k+1)^3); \\ _Michel Marcus_, Nov 14 2015
%Y A222636 Column k=3 of A383049.
%Y A222636 Cf. A223901.
%K A222636 sign
%O A222636 0,2
%A A222636 _Takao Komatsu_, Mar 28 2013