A129825 a(n) = n!*Bernoulli(n-1), n > 2; a(0)=0, a(1)=1, a(2)=1.

0, 1, 1, 1, 0, -4, 0, 120, 0, -12096, 0, 3024000, 0, -1576143360, 0, 1525620096000, 0, -2522591034163200, 0, 6686974460694528000, 0, -27033456071346536448000, 0, 160078872315904478576640000, 0, -1342964491649083924630732800000, 0, 15522270327163593186886877184000000, 0

Offset: 0

Author

Paul Curtz, Jun 03 2007

Keywords

• sign

Comments

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).

The sequence is then defined by a(n) = n!*G(n).

The first differences are 1, 0, 0, -1, -4, 4, 120, -120, -12096, ...

The 2nd differences are -1, 0, -1, -3, 8, 116, -240, -11976, 24192, 3011904, ...

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A001332, A027641, A027642, A129716, A129814, A129826.

Equals second left hand column of A161739 (RSEG2 triangle).

Other left hand columns are A161742 and A161743.

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].

Programs

Magma

[n le 2 select Floor((n+1)/2) else Factorial(n)*Bernoulli(n-1): n in [0..40]]; // G. C. Greubel, Apr 26 2024

Maple

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

Mathematica

a[n_] := n!*BernoulliB[n-1]; a[0]=0; a[2]=1; Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Mar 04 2013 *)

SageMath

[(n+1)//2 if n <3 else factorial(n)*bernoulli(n-1) for n in range(41)] # G. C. Greubel, Apr 26 2024

Formula

From Johannes W. Meijer, Jun 18 2009: (Start)

a(n) = Sum_{k=1..n} (-1)^(k+1)*(n!/k)*S2(n, k)*(k-1)!.

a(n) = Sum_{k=0..n-1} ((-1)^k/(k!*(k+1)!))*n!*A028246(n, k+1) *A008955(k, k). (End)

a(n) = A129814(n-1) for n > 2. - Georg Fischer, Oct 07 2018

Extensions

Edited by R. J. Mathar, May 21 2009