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A222748 Poly-Cauchy numbers c_n^(-4).

%I A222748 #50 Apr 16 2025 09:02:26
%S A222748 1,16,65,45,-116,340,-1240,5480,-28464,169248,-1125840,8197680,
%T A222748 -63806016,514314240,-4058967744,26952984000,-37203513984,
%U A222748 -4251686488704,140692872720384,-3560137793538048,84004474130786304,-1955196907518928896,45927815909901004800
%N A222748 Poly-Cauchy numbers c_n^(-4).
%C A222748 Definition of poly-Cauchy numbers in A222627.
%H A222748 Vincenzo Librandi, <a href="/A222748/b222748.txt">Table of n, a(n) for n = 0..300</a>
%H A222748 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 A222748 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 A222748 Takao Komatsu, <a href="http://doi.org/10.2206/kyushujm.67.143">Poly-Cauchy numbers</a>, Kyushu J. Math. 67 (2013), 143-153.
%H A222748 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 A222748 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 A222748 Wikipedia, <a href="http://en.wikipedia.org/wiki/Stirling_transform">Stirling transform</a>
%F A222748 a(n) = Sum_{k=0..n} Stirling1(n,k) * (k+1)^4.
%F A222748 From _Seiichi Manyama_, Apr 14 2025: (Start)
%F A222748 E.g.f.: Sum_{k>=0} (k+1)^4 * log(1+x)^k / k!.
%F A222748 E.g.f.: (1+x) * Sum_{k=0..4} Stirling2(5,k+1) * log(1+x)^k. (End)
%t A222748 Table[Sum[StirlingS1[n, k] (k + 1)^4, {k, 0, n}], {n, 0, 25}]
%o A222748 (Magma) [&+[StirlingFirst(n,k)*(k+1)^4: k in [0..n]]: n in [0..25]]; // _Bruno Berselli_, Mar 28 2013
%o A222748 (PARI) a(n) = sum(k=0, n, stirling(n, k, 1)*(k+1)^4); \\ _Michel Marcus_, Nov 14 2015
%Y A222748 Column k=4 of A383049.
%K A222748 sign
%O A222748 0,2
%A A222748 _Takao Komatsu_, Mar 28 2013