This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A223904 #34 Apr 15 2025 08:26:09
%S A223904 1,-32,275,-1817,12134,-87784,699894,-6158058,59566464,-630057696,
%T A223904 7246806720,-90151868160,1207028135520,-17314992935040,
%U A223904 265048030579680,-4313510679824160,74387763047472000,-1355291635314213120,26016022725597866880,-524865277479851360640,11103724030717930095360
%N A223904 Poly-Cauchy numbers of the second kind hat c_n^(-5).
%C A223904 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 A223904 Vincenzo Librandi, <a href="/A223904/b223904.txt">Table of n, a(n) for n = 0..300</a>
%H A223904 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 A223904 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 A223904 Takao Komatsu, <a href="http://doi.org/10.2206/kyushujm.67.143">Poly-Cauchy numbers</a>, Kyushu J. Math. 67 (2013), 143-153.
%F A223904 a(n) = Sum_{k=0..n} (-1)^k * (k+1)^5 * Stirling1(n,k).
%F A223904 From _Seiichi Manyama_, Apr 15 2025: (Start)
%F A223904 E.g.f.: Sum_{k>=0} (k+1)^5 * (-log(1+x))^k / k!.
%F A223904 E.g.f.: (1/(1+x)) * Sum_{k=0..5} Stirling2(6,k+1) * (-log(1+x))^k.
%F A223904 a(n) = (-1)^n * Sum_{k=0..5} k! * Stirling2(6,k+1) * |Stirling1(n+1,k+1)|. (End)
%t A223904 Table[Sum[StirlingS1[n, k] (-1)^k (k + 1)^5, {k, 0, n}], {n, 0, 30}]
%o A223904 (Magma) [&+[StirlingFirst(n, k)*(-1)^k*(k+1)^5: k in [0..n]]: n in [0..23]]; // _Vincenzo Librandi_, Aug 03 2013
%o A223904 (PARI) a(n) = sum(k=0, n, (-1)^k*stirling(n, k, 1)*(k+1)^5); \\ _Michel Marcus_, Nov 14 2015
%Y A223904 Cf. A081046, A223899, A223901, A223902.
%Y A223904 Cf. A223023.
%K A223904 sign
%O A223904 0,2
%A A223904 _Takao Komatsu_, Mar 29 2013