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A223852 Poly-Cauchy numbers c_5^(-n).

%I A223852 #34 Sep 02 2025 21:03:05
%S A223852 -6,-8,48,340,984,-1148,-34152,-254780,-1250376,-3417788,12508248,
%T A223852 296104900,3122953464,26485493572,201873508248,1443404093380,
%U A223852 9892106472504,65798800964932,428187502981848,2740792716574660,17321987718906744,108394003491348292
%N A223852 Poly-Cauchy numbers c_5^(-n).
%C A223852 Definition of poly-Cauchy numbers in A222627.
%H A223852 Vincenzo Librandi, <a href="/A223852/b223852.txt">Table of n, a(n) for n = 1..300</a>
%H A223852 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 A223852 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 A223852 Takao Komatsu, <a href="http://doi.org/10.2206/kyushujm.67.143">Poly-Cauchy numbers</a>, Kyushu J. Math. 67 (2013), 143-153.
%H A223852 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 A223852 Wikipedia, <a href="http://en.wikipedia.org/wiki/Stirling_transform">Stirling transform</a>
%H A223852 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (20,-155,580,-1044,720).
%F A223852 a(n) = Sum_{k=0..5} Stirling1(5,k)*(k+1)^n.
%F A223852 Empirical g.f.: -2*x*(810*x^3 - 361*x^2 + 56*x - 3) / ((2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)). - _Colin Barker_, Mar 31 2013
%t A223852 Table[Sum[StirlingS1[5, k] (k + 1)^n, {k, 0, 5}], {n, 25}]
%o A223852 (Magma) [&+[StirlingFirst(5,k)*(k+1)^n: k in [0..5]]: n in [1..25]]; // _Bruno Berselli_, Mar 28 2013
%o A223852 (PARI) a(n) = sum(k=0, 5, stirling(5, k, 1)*(k+1)^n); \\ _Michel Marcus_, Nov 14 2015
%K A223852 sign,changed
%O A223852 1,1
%A A223852 _Takao Komatsu_, Mar 28 2013