A001332 -id:A001332 - cp's OEIS frontend

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

Paul Curtz, Jun 03 2007

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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, ...

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

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].

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

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

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

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

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

Edited by R. J. Mathar, May 21 2009

A129814 a(n) = Bernoulli(n) * (n+1)!.

Original entry on oeis.org

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

Offset: 0

Paul Curtz, May 20 2007

From Peter Luschny, Apr 21 2009: (Start)
Reading A137777 and A159749 as a triangular sequence:
2*a(n) = A137777(n, 0) for n > 0.
2*a(n) = (-1)^n*A159749(n, 0) for n >= 0. (End)

Eric Weisstein's World of Mathematics, Bernoulli Number
Eric Weisstein's World of Mathematics, Polygamma Function

Cf. A001332.

[Bernoulli(n) * Factorial(n+1): n in [0..100]]; // Vincenzo Librandi, Mar 29 2011

Table[BernoulliB[n](n+1)!,{n,0,30}] (* Harvey P. Dale, Jan 18 2013 *)
Table[SeriesCoefficient[-2 x - PolyGamma[2, 1/x] / x^2, {x, 0, n}, Assumptions -> x > 0] n!, {n, 0, 30}] (* Vladimir Reshetnikov, Apr 24 2013 *)

{for(n=0, 25, print1(bernfrac(n)*(n+1)!, ","))}

{a(n) = if( n<0, 0, (n + 1)! * bernfrac( n))} /* Michael Somos, Mar 29 2011 */

a(2*n) = A001332(n).

E.g.f.: -2 x - psi_2(1/x) / x^2, where psi_n(z) is the polygamma function, psi_n(z) = (d/dz)^{n+1} log(Gamma(z)). - Vladimir Reshetnikov, Apr 24 2013

Edited and extended by Klaus Brockhaus, May 28 2007

A263445 a(n) = (2n+1)(n+1)!Bernoulli(2n).

Original entry on oeis.org

1, 1, -1, 4, -36, 600, -16584, 705600, -43751232, 3790108800, -443539877760, 68218849036800, -13478425925184000, 3355402067989171200, -1035218714714606822400, 390189256983139461120000, -177430554756972746695065600, 96269372301568677170319360000

Offset: 0

Vladimir Reshetnikov, Oct 18 2015

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Robert Israel, Table of n, a(n) for n = 0..210
Index entries for sequences related to Bernoulli numbers.

Bernoulli numbers are A000367/A002445. Cf. A004193, A001332, A000182, A001469.

seq((2*n+1)*(n+1)!*bernoulli(2*n), n=0..50); # Robert Israel, Oct 18 2015

Table[(2n + 1) (n + 1)! BernoulliB[2n], {n, 0, 17}]

vector(30, n, n--; (2*n+1)*(n+1)!*bernfrac(2*n)) \\ Altug Alkan, Oct 18 2015

from math import factorial
from sympy import bernoulli
def A263445(n): return (2*n+1)*factorial(n+1)*bernoulli(2*n) # Chai Wah Wu, May 18 2022

a(n) = (2n+1)*(n+1)!*Bernoulli(2n).

a(n) ~ (-1)^(n+1)*8*sqrt(2)*n^3*(n/e)^(3*n)*Pi^(1-2*n). - Vladimir Reshetnikov, Sep 05 2016

A168136 a(n) = Bernoulli(2n)*(2n+1)!/n!.

Original entry on oeis.org

1, 1, -2, 20, -504, 25200, -2189088, 302702400, -62564261760, 18427508985600, -7449695786856960, 4010313259477324800, -2803674333549374208000, 2492728196309155284480000, -2768630339381333070099456000

Offset: 0

Franklin T. Adams-Watters, Nov 19 2009

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Given by alternating sums of coefficients of polynomials spawned by sums of binomial(n,k)^2*k^(2m). - John M. Campbell, Nov 30 2011

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

Cf. A001332, A004193.

[Bernoulli(2*n)*Factorial(2*n+1)/Factorial(n): n in [0..20]]; // Vincenzo Librandi, Jul 14 2016

Table[BernoulliB[2*n]*(2*n+1)!/n!, {n, 0, 20}]

PARI
```
a(n)=bernfrac(2*n)*(2*n+1)!/n!
```

A347425 a(n) = Bernoulli(2n) (2n+1)! if 2n+1 is a prime, otherwise a(n) = Bernoulli(2n) (2*n)!.

Original entry on oeis.org

1, 1, -4, 120, -1344, 3024000, -1576143360, 101708006400, -2522591034163200, 6686974460694528000, -1287307431968882688000, 160078872315904478576640000, -53718579665963356985229312000, 574898901006059006921736192000000, -241364461951740682229320388129587200000

Offset: 0

Ilya Gutkovskiy, Sep 01 2021

			Bernoulli(2*n) * (2*n)! = [ 1, 1/3, -4/5, 120/7, -1344, 3024000/11, -1576143360/13, 101708006400, -2522591034163200/17, 6686974460694528000/19, ... ].

Index entries for sequences related to Bernoulli numbers

Cf. A000367, A001332, A002431, A002445, A036278, A128059.

a[n_] := If[PrimeQ[2 n + 1], BernoulliB[2 n] (2 n + 1)!, BernoulliB[2 n] (2 n)!]; Table[a[n], {n, 0, 14}]
Table[Numerator[BernoulliB[2 n] (2 n)!], {n, 0, 14}]
Table[Numerator[(2 n)!^2 SeriesCoefficient[x Coth[x/2]/2, {x, 0, 2 n}]], {n, 0, 14}]
b[0] = 1; b[n_] := b[n] = -Sum[Binomial[n, k]^2 k! b[n - k]/(k + 1), {k, 1, n}]; a[n_] := Numerator[b[2 n]]; Table[a[n], {n, 0, 14}]

a(n) = numerator(bernfrac(2*n)*(2*n)!); \\ Michel Marcus, Sep 01 2021

a(n) is the numerator of Bernoulli(2*n) * (2*n)! (for denominators see A128059).
a(n) is the numerator of (2*n)!^2 * [x^(2*n)] x * coth(x/2) / 2.
a(n) is the numerator of b(2*n) where b(n) = -Sum_{k=1..n} binomial(n,k)^2 * k! * b(n-k) / (k+1), b(0) = 1.

cp's OEIS Frontend

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

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A129814 a(n) = Bernoulli(n) * (n+1)!.

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A263445 a(n) = (2n+1)*(n+1)!*Bernoulli(2n).

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A168136 a(n) = Bernoulli(2n)*(2n+1)!/n!.

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A347425 a(n) = Bernoulli(2*n) * (2*n+1)! if 2*n+1 is a prime, otherwise a(n) = Bernoulli(2*n) * (2*n)!.

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A263445 a(n) = (2n+1)(n+1)!Bernoulli(2n).

A347425 a(n) = Bernoulli(2n) (2n+1)! if 2n+1 is a prime, otherwise a(n) = Bernoulli(2n) (2*n)!.