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Poly-Cauchy numbers of the second kind -hat c_3^(-n).
1

%I #27 Feb 05 2023 12:28:14

%S 17,51,161,531,1817,6411,23201,85731,322217,1227771,4729841,18379731,

%T 71908217,282817131,1116854081,4424238531,17567551817,69882262491,

%U 278365739921,1109974078131,4429431765017,17686337611851,70651190491361,282322298874531

%N Poly-Cauchy numbers of the second kind -hat c_3^(-n).

%C 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 Vincenzo Librandi, <a href="/A223906/b223906.txt">Table of n, a(n) for n = 1..1000</a>

%H 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 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 Takao Komatsu, <a href="http://doi.org/10.2206/kyushujm.67.143">Poly-Cauchy numbers</a>, Kyushu J. Math. 67 (2013), 143-153.

%F Conjecture: a(n) = 2^(1+n)+3^(1+n)+4^n. G.f.: -x*(144*x^2-102*x+17) / ((2*x-1)*(3*x-1)*(4*x-1)). - _Colin Barker_, Mar 31 2013

%t Table[-Sum[StirlingS1[3, k] (-1)^k (k + 1)^n, {k, 0, 3}], {n, 30}]

%o (PARI) a(n) = -sum(k=0, 3, (-1)^k*stirling(3, k, 1)*(k+1)^n); \\ _Michel Marcus_, Nov 14 2015

%Y Cf. A223173.

%K nonn,easy

%O 1,1

%A _Takao Komatsu_, Mar 29 2013