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A222627 Poly-Cauchy numbers c_n^(-2) (for definition see Comments lines).

%I A222627 #78 Apr 16 2025 09:02:13
%S A222627 1,4,5,-3,4,-8,20,-52,72,936,-17568,238752,-3113280,41503680,
%T A222627 -577877760,8470414080,-131039838720,2139954163200,-36854615347200,
%U A222627 668374040678400,-12742107588403200,254904791591116800,-5341386032640000000,117034910701793280000
%N A222627 Poly-Cauchy numbers c_n^(-2) (for definition see Comments lines).
%C A222627 The definition of poly-Cauchy numbers is given in Theorem 1 of the paper Poly-Cauchy numbers (see Links lines).
%C A222627 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 A222627 Vincenzo Librandi, <a href="/A222627/b222627.txt">Table of n, a(n) for n = 0..200</a>
%H A222627 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 A222627 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 A222627 Takao Komatsu, <a href="http://doi.org/10.2206/kyushujm.67.143">Poly-Cauchy numbers</a>, Kyushu J. Math. 67 (2013), 143-153.
%H A222627 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 A222627 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 A222627 Wikipedia, <a href="http://en.wikipedia.org/wiki/Stirling_transform">Stirling transform</a>
%F A222627 a(n) = Sum_{k=0..n} Stirling1(n,k) * (k+1)^2.
%F A222627 E.g.f.: (1 + x) * (1 + log(1 + x) * (3 + log(1 + x))). - _Ilya Gutkovskiy_, Aug 09 2021
%F A222627 E.g.f.: Sum_{k>=0} (k+1)^2 * log(1+x)^k / k!. - _Seiichi Manyama_, Apr 14 2025
%t A222627 Table[Sum[StirlingS1[n, k]*(k + 1)^2, {k, 0, n}], {n, 0, 25}]
%o A222627 (Magma) [&+[StirlingFirst(n,k)*(k+1)^2: k in [0..n]]: n in [0..25]]; // _Bruno Berselli_, Mar 28 2013
%o A222627 (PARI) a(n) = sum(k=0, n, stirling(n, k, 1)*(k+1)^2); \\ _Michel Marcus_, Nov 14 2015
%Y A222627 Column k=2 of A383049.
%Y A222627 Cf. A006233.
%K A222627 sign
%O A222627 0,2
%A A222627 _Takao Komatsu_, Mar 28 2013