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A223908 Poly-Cauchy numbers of the second kind -hat c_5^(-n).

%I A223908 #28 Sep 02 2025 21:07:41
%S A223908 394,1392,5248,20940,87784,384252,1747048,8213820,39780424,197799612,
%T A223908 1006785448,5232061500,27696448264,149034102972,813659961448,
%U A223908 4499466577980,25163809551304,142131488326332,809773455691048,4648490027827260,26859776918289544
%N A223908 Poly-Cauchy numbers of the second kind -hat c_5^(-n).
%C A223908 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 A223908 Vincenzo Librandi, <a href="/A223908/b223908.txt">Table of n, a(n) for n = 1..1000</a>
%H A223908 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 A223908 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 A223908 Takao Komatsu, <a href="http://doi.org/10.2206/kyushujm.67.143">Poly-Cauchy numbers</a>, Kyushu J. Math. 67 (2013), 143-153.
%H A223908 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (20,-155,580,-1044,720).
%F A223908 Empirical g.f.: -2*x*(43200*x^4-48390*x^3+19239*x^2-3244*x+197) / ((2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)). - _Colin Barker_, Mar 31 2013
%t A223908 Table[-Sum[StirlingS1[5, k] (-1)^k (k + 1)^n, {k, 0, 5}], {n, 30}]
%o A223908 (PARI) a(n) = -sum(k=0, 5, (-1)^k*stirling(5, k, 1)*(k+1)^n); \\ _Michel Marcus_, Nov 14 2015
%Y A223908 Cf. A223852.
%K A223908 nonn,easy,changed
%O A223908 1,1
%A A223908 _Takao Komatsu_, Mar 29 2013