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A223023 Poly-Cauchy numbers c_n^(-5).

%I A223023 #58 Apr 16 2025 09:02:17
%S A223023 1,32,211,359,-538,984,-1866,1110,32640,-449760,5035200,-55896960,
%T A223023 646005600,-7896549120,102604234080,-1418189492640,20828546505600,
%U A223023 -324419255412480,5346952977432960,-93035974518691200,1705088403923592960,-32842738382065931520
%N A223023 Poly-Cauchy numbers c_n^(-5).
%C A223023 Definition of poly-Cauchy numbers in A222627.
%H A223023 Vincenzo Librandi, <a href="/A223023/b223023.txt">Table of n, a(n) for n = 0..300</a>
%H A223023 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 A223023 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 A223023 Takao Komatsu, <a href="http://doi.org/10.2206/kyushujm.67.143">Poly-Cauchy numbers</a>, Kyushu J. Math. 67 (2013), 143-153.
%H A223023 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 A223023 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 A223023 Wikipedia, <a href="http://en.wikipedia.org/wiki/Stirling_transform">Stirling transform</a>
%F A223023 a(n) = Sum_{k=0..n} Stirling1(n,k) * (k+1)^5.
%F A223023 From _Seiichi Manyama_, Apr 14 2025: (Start)
%F A223023 E.g.f.: Sum_{k>=0} (k+1)^5 * log(1+x)^k / k!.
%F A223023 E.g.f.: (1+x) * Sum_{k=0..5} Stirling2(6,k+1) * log(1+x)^k. (End)
%t A223023 Table[Sum[StirlingS1[n, k] (k + 1)^5, {k, 0, n}], {n, 0, 25}]
%o A223023 (Magma) [&+[StirlingFirst(n,k)*(k+1)^5: k in [0..n]]: n in [0..25]]; // _Bruno Berselli_, Mar 28 2013
%o A223023 (PARI) a(n) = sum(k=0, n, stirling(n, k, 1)*(k+1)^5); \\ _Michel Marcus_, Nov 14 2015
%Y A223023 Column k=5 of A383049.
%K A223023 sign
%O A223023 0,2
%A A223023 _Takao Komatsu_, Mar 28 2013