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 A129825 #32 Apr 28 2024 11:37:47
%S A129825 0,1,1,1,0,-4,0,120,0,-12096,0,3024000,0,-1576143360,0,1525620096000,
%T A129825 0,-2522591034163200,0,6686974460694528000,0,-27033456071346536448000,
%U A129825 0,160078872315904478576640000,0,-1342964491649083924630732800000,0,15522270327163593186886877184000000,0
%N A129825 a(n) = n!*Bernoulli(n-1), n > 2; a(0)=0, a(1)=1, a(2)=1.
%C A129825 Define "conjugated" Bernoulli numbers G(n) via G(0)=0, G(1)=B(0)=1, G(2)=-B(1)=1/2, G(n+1)=B(n), where B(n)=A027641(n)/A027642(n).
%C A129825 The sequence is then defined by a(n) = n!*G(n).
%C A129825 The first differences are 1, 0, 0, -1, -4, 4, 120, -120, -12096, ...
%C A129825 The 2nd differences are -1, 0, -1, -3, 8, 116, -240, -11976, 24192, 3011904, ...
%H A129825 Vincenzo Librandi, <a href="/A129825/b129825.txt">Table of n, a(n) for n = 0..100</a>
%F A129825 From _Johannes W. Meijer_, Jun 18 2009: (Start)
%F A129825 a(n) = Sum_{k=1..n} (-1)^(k+1)*(n!/k)*S2(n, k)*(k-1)!.
%F A129825 a(n) = Sum_{k=0..n-1} ((-1)^k/(k!*(k+1)!))*n!*A028246(n, k+1) *A008955(k, k). (End)
%F A129825 a(n) = A129814(n-1) for n > 2. - _Georg Fischer_, Oct 07 2018
%p A129825 A129825 := proc(n) if n <= 1 then n; elif n = 2 then 1; else n!*bernoulli(n-1) ; fi; end: # _R. J. Mathar_, May 21 2009
%t A129825 a[n_] := n!*BernoulliB[n-1]; a[0]=0; a[2]=1; Table[a[n], {n, 0, 28}] (* _Jean-François Alcover_, Mar 04 2013 *)
%o A129825 (Magma)
%o A129825 [n le 2 select Floor((n+1)/2) else Factorial(n)*Bernoulli(n-1): n in [0..40]]; // _G. C. Greubel_, Apr 26 2024
%o A129825 (SageMath)
%o A129825 [(n+1)//2 if n <3 else factorial(n)*bernoulli(n-1) for n in range(41)] # _G. C. Greubel_, Apr 26 2024
%Y A129825 Cf. A001332, A027641, A027642, A129716, A129814, A129826.
%Y A129825 Equals second left hand column of A161739 (RSEG2 triangle).
%Y A129825 Other left hand columns are A161742 and A161743.
%Y A129825 Cf. A094310 [T(n,k) = n!/k], A008277 [S2(n,k); Stirling numbers of the second kind], A028246 [Worpitzky's triangle] and A008955 [CFN triangle].
%K A129825 sign
%O A129825 0,6
%A A129825 _Paul Curtz_, Jun 03 2007
%E A129825 Edited by _R. J. Mathar_, May 21 2009