cp's OEIS Frontend

A223902 Poly-Cauchy numbers of the second kind hat c_n^(-4).

%I A223902 #32 Apr 15 2025 08:26:13
%S A223902 1,-16,97,-531,3148,-20940,156680,-1310840,12166096,-124281120,
%T A223902 1387313520,-16813355280,219967479744,-3090914335104,46439677053120,
%U A223902 -743069262651840,12616998421804416,-226608929801923968,4292762009479969536,-85545808260446050560,1789078468694176410624
%N A223902 Poly-Cauchy numbers of the second kind hat c_n^(-4).
%C A223902 The poly-Cauchy numbers of the second kind hat c_n^k can be expressed in terms of the (unsigned) Stirling numbers of the first kind: hat c_n^(k) = (-1)^n*sum(abs(stirling1(n,m))/(m+1)^k, m=0..n).
%H A223902 Vincenzo Librandi, <a href="/A223902/b223902.txt">Table of n, a(n) for n = 0..300</a>
%H A223902 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 A223902 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), p. 42-53.
%H A223902 Takao Komatsu, <a href="http://doi.org/10.2206/kyushujm.67.143">Poly-Cauchy numbers</a>, Kyushu J. Math. 67 (2013), 143-153.
%F A223902 a(n) = Sum_{k=0..n} (-1)^k * (k+1)^4 * Stirling1(n,k).
%F A223902 From _Seiichi Manyama_, Apr 15 2025: (Start)
%F A223902 E.g.f.: Sum_{k>=0} (k+1)^4 * (-log(1+x))^k / k!.
%F A223902 E.g.f.: (1/(1+x)) * Sum_{k=0..4} Stirling2(5,k+1) * (-log(1+x))^k.
%F A223902 a(n) = (-1)^n * Sum_{k=0..4} k! * Stirling2(5,k+1) * |Stirling1(n+1,k+1)|. (End)
%t A223902 Table[Sum[StirlingS1[n, k] (-1)^k (k + 1)^4, {k, 0, n}], {n, 0, 30}]
%o A223902 (PARI) a(n) = sum(k=0, n, stirling(n, k, 1)*(-1)^k*(k+1)^4); \\ _Michel Marcus_, Nov 14 2015
%Y A223902 Cf. A081046, A223899, A223901, A223904.
%Y A223902 Cf. A222748.
%K A223902 sign
%O A223902 0,2
%A A223902 _Takao Komatsu_, Mar 29 2013