A141056 -id:A141056 - cp's OEIS frontend

A164020 Denominators of Bernoulli numbers interleaved with even numbers.

1, 2, 6, 4, 30, 6, 42, 8, 30, 10, 66, 12, 2730, 14, 6, 16, 510, 18, 798, 20, 330, 22, 138, 24, 2730, 26, 6, 28, 870, 30, 14322, 32, 510, 34, 6, 36, 1919190, 38, 6, 40, 13530, 42, 1806, 44, 690, 46, 282, 48, 46410, 50, 66, 52, 1590, 54, 798, 56, 870, 58, 354, 60, 56786730

Offset: 0

Paul Curtz, Aug 08 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Bernoulli Number

Cf. A027642, A141056, A160014.

[IsEven(n) select Denominator(Bernoulli(n)) else n+1: n in [0..100]]; // Vincenzo Librandi, Sep 08 2017

a[n_]:=If[OddQ[n], n+1, BernoulliB[n] // Denominator]; Table[a[n], {n, 0, 60}](* Jean-François Alcover, Dec 29 2012 *)
With[{nn=60},Riffle[Denominator[BernoulliB[Range[0,nn,2]]],Range[2,nn,2]]] (* Harvey P. Dale, Jul 18 2015 *)

a(2*n) = A002445(n).
a(2*n+1) = 2*(n+1).
a(n) divides A057643(n). Franklin T. Adams-Watters, Aug 03 2012

Extended by R. J. Mathar, Sep 23 2009

A185424 Numerators of generalized Bernoulli numbers associated with the zigzag numbers A000111.

Original entry on oeis.org

1, -1, 1, -1, 19, -5, 253, -61, 3319, -1385, 222557, -50521, 422152729, -2702765, 59833795, -199360981, 439264083023, -19391512145, 76632373664299, -2404879675441, 4432283799315809, -370371188237525

Offset: 0

Peter Bala, Feb 18 2011

DEFINITION
Let E(t) = sec(t)+tan(t) denote the generating function for the zigzag numbers A000111. The zigzag Bernoulli numbers, denoted ZB(n), are defined by means of the generating function
(1)... log E(t)/(E(t)-1) = sum {n = 0..inf} ZB(n)*t^n/n!.
Notice that if we were to take E(t) equal to exp(t) then (1) would be the defining function for the classical Bernoulli numbers B_n. The first few even-indexed values of ZB(n) are
....n..|..0...2.....4.......6........8..........10...........12....
===================================================================
.ZB(n).|..1..1/6..19/30..253/42..3319/30..222557/66..422152729/2730
while the odd-indexed values begin
....n..|. ..1......3......5.......7........9.........11..
=========================================================
.ZB(n).|. -1/2...-1/2...-5/2...-61/2...-1385/2...-50521/2
The present sequence gives the numerators of the zigzag Bernoulli numbers. It is not difficult to show that the odd-indexed value ZB(2*n+1) equals -1/2*A000364(n). The numerators of the even-indexed values ZB(2*n) are shown separately in A185425.
VON STAUDT-CLAUSEN THEOREM
The following analog of the von Staudt-Clausen theorem holds:
(2)... ZB(2*n) + 1/2 + S(1) + (-1)^(n+1)*S(3) equals an integer, where
... S(1) = sum {prime p, p = 1 (mod 4), p-1|2*n} 1/p,
... S(3) = sum {prime p, p = 3 (mod 4), p-1|2*n} 1/p.
For example,
(3)... ZB(12) + 1/2 + (1/5+1/13) - (1/3+1/7) = 154635.
Further examples are given below.

			Examples of von Staudt and Clausen's theorem for ZB(2*n):
ZB(2) = 1/6 = 1 - 1/2 - 1/3;
ZB(4) = 19/30 = 1 - 1/2 + 1/3 - 1/5;
ZB(6) = 253/42 = 7 - 1/2 - 1/3 - 1/7;
ZB(8) = 3319/30 = 111 - 1/2 + 1/3 - 1/5;
ZB(10) = 222557/66 = 3373 - 1/2 - 1/3 - 1/11.

Cf. A000111, A027641, A027642, A147309, A185425.

Sequence of denominators is A141056.

#A185424
a:= n-> numer((-1)^(n*(n-1)/2)*add(binomial(n,k)/(k+1)* bernoulli(n-k) *euler(k), k = 0..n)):
seq(a(n), n = 0..20);

Numerator[ Range[0, 30]! CoefficientList[ Series[Log(Sec[x]+Tan[x])/(Sec[x] +Tan[x] - 1), {x, 0, 30}], x]]

SEQUENCE ENTRIES
a(n) = numerator of the rational number ZB(n) where
(1)... ZB(n) = (-1)^(n*(n-1)/2)*sum {k = 0..n} binomial(n,k)/(k+1)* Bernoulli(n- k)*Euler(k).
For odd indices this simplifies to
(2)... ZB(2*n+1) = (-1)^n*Euler(2*n)/2, where Euler(2*n) = A028296(n).
For even indices we have
(3)... ZB(2*n) = (-1)^n*sum {k = 0..n} binomial(2*n,2*k)/(2*k+1)* Bernoulli(2*n- 2*k)*Euler(2*k).
GENERATING FUNCTION
E.g.f:
(4)... log(sec(t)+tan(t))/(sec(t)+tan(t)-1) =
   1 -1/2*t +1/6*t^2/2! -1/2*t^3/3! + ....
RELATION WITH ZIGZAG POLYNOMIALS OF A147309
The classical Bernoulli numbers B_n are given by the double sum
(5)... B_n = sum {k=0..n} sum {j=0..k} (-1)^j*binomial(k,j)*j^n/(k+1).
The corresponding formula for the zigzag Bernoulli numbers is
(6)... ZB(n) = sum {k=0..n} sum {j=0..k}(-1)^j*binomial(k,j)*Z(n,j)/(k+1), where Z(n,x) is a zigzag polynomial as defined in A147309. Umbrally, we can express this as
(7)... ZB(n) = Z(n,B), where on the lhs the understanding is that in the expansion of the zigzag polynomial Z(n,x) a term such as c_k*x^k is to be replaced with c_k*B_k. For example, Z(6,x) = 40*x^2+20*x^4+x^6 and so ZB(6) = 40*B_2+20*B_4+B_6 = 40*(1/6)+20*(-1/30)+(1/42) = 253/42.

A226040 a(n) = product{ p prime such that p divides n + 1 and p - 1 does not divide n }.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 7, 5, 1, 1, 3, 1, 5, 7, 11, 1, 3, 1, 13, 1, 7, 1, 15, 1, 1, 11, 17, 35, 3, 1, 19, 13, 5, 1, 21, 1, 11, 1, 23, 1, 3, 1, 5, 17, 13, 1, 3, 55, 7, 19, 29, 1, 15, 1, 31, 7, 1, 13, 33, 1, 17, 23, 35, 1, 3, 1, 37, 5, 19, 77, 39

Offset: 0

Peter Luschny, May 26 2013

nonn

			a(41) = 21 = 3*7 = product({2,3,7} setminus {2}).

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Peter Luschny, Generalized Bernoulli numbers.

Cf. A160014, A226038, A226039, A225481, A141056.

s:= (p, n) -> ((n+1) mod p = 0) and (n mod (p-1) <> 0);
A226040 := n -> mul(z, z = select(p->s(p,n), select('isprime', [$2..n])));
seq(A226040(n), n=0..77);

a[n_] := Times @@ Select[ FactorInteger[n+1][[All, 1]], !Divisible[n, #-1] &]; a[0] = 1; Table[a[n], {n, 0, 77}] (* Jean-François Alcover, Jun 27 2013, after Maple *)

a(n)=my(f=factor(n+1)[,1],s=1);prod(i=1,#f,if(n%(f[i]-1),f[i],1)) \\ Charles R Greathouse IV, Jun 27 2013

def A226040(n):
    F = filter(lambda p: ((n+1) % p == 0) and (n % (p-1)), primes(n))
    return mul(F)
[A226040(n) for n in (0..77)]

a(n) = A225481(n) / A141056(n).

A225825 a(2n)=A001896(n). a(2n+1)=(-1)^n*A110501(n+1).

Original entry on oeis.org

1, 1, -1, -1, 7, 3, -31, -17, 127, 155, -2555, -2073, 1414477, 38227, -57337, -929569, 118518239, 28820619, -5749691557, -1109652905, 91546277357, 51943281731, -1792042792463, -2905151042481, 1982765468311237, 191329672483963, -286994504449393, -14655626154768697, 3187598676787461083, 1291885088448017715, -4625594554880206790555

Offset: 0

Paul Curtz, Jul 30 2013

sign

a(n) is the numerators of numbers derived from Bernoulli and Genocchi numbers. The denominators b(n) are the Clausen numbers A141056.
The numbers are
BERGEN(n) = 1, 1/2, -1/6, -1/2, 7/30, 3/2, -31/42, -17/2, 127/30, 155/2,..
Difference table:
1,       1/2,     -1/6,     -1/2,      7/30,        3/2,    -31/42,...
-1/2,   -2/3,     -1/3,    11/15,     19/15,     -47/21,   -163/21,...
-1/6,    1/3,    16/15,     8/15,  -368/105,    -116/21,  2152/105,...
1/2,   11/15,    -8/15, -424/105,  -212/105,   2732/105,  4204/105,...
7/30, -19/15, -368/195,  212/105,  2944/105,   1472/105,...
-3/2, -47/21,   116/21, 2732/105, -1472/105, -70240/231, -35120/231,... .
a(n) is an autosequence. Its inverse binomial transform is the sequence signed. Its main diagonal is the double of the first upper diagonal.
a(n) is divisible by A051716(n+1).
Denominators of the main diagonal: A181131(n). Checked by Jean-François Alcover for the first 25 terms.
The numerators of the main diagonal:
1, -2, 16, -424, 2944, -70240, 70873856, -212648576, 98650550272,...
(thanks to Jean-François Alcover) are divisible by 2^n.

Cf. A083420.

A225825 := proc(n)
    local nhalf ;
    nhalf := floor(n/2) ;
    if type(n,'even') then
        A001896(nhalf) ;
    else
        (-1)^nhalf*A110501(nhalf+1) ;
    end if;
end proc; # R. J. Mathar, Oct 28 2013

a[0] = 1; a[n_] := Numerator[BernoulliB[n, 1/2] - (n+1)*EulerE[n, 0]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 01 2013 *)

c(n)=(0 followed by -A036968(n+1)) = 0, 1, 0, -1, 0, 3,... .

a(n) = A157779(n) + c(n).

More terms from Jean-François Alcover, Aug 01 2013

Definition corrected by R. J. Mathar, Oct 28 2013

A230324 a(n) = A226158(n) - 2*A226158(n+1).

Original entry on oeis.org

2, 1, -1, -2, 1, 6, -3, -34, 17, 310, -155, -4146, 2073, 76454, -38227, -1859138, 929569, 57641238, -28820619, -2219305810, 1109652905, 103886563462, -51943281731, -5810302084962, 2905151042481, 382659344967926

Offset: 0

Paul Curtz, Oct 16 2013

sign

The array A(n,k) = A(n-1,k+1) - A(n-1,k) of the sequence in the first row and higher-order sequences in followup rows starts:
   2,  1,  -1,   -2,     1,     6,    -3, ...
  -1, -2,  -1,    3,     5,    -9,   -31, ...
  -1,  1,   4,    2,   -14,   -22,    82, ...
   2,  3,  -2,  -16,    -8,   104,   160, ...
   1, -5, -14,    8,   112,    56, -1160, ...
  -6, -9,  22,  104,   -56, -1216,  -608, ...
  -3, 31,  82, -160, -1160,   608, 18880, ...
etc.
a(n) is an autosequence: Its inverse binomial transform is the sequence (up to a sign), which means top row and left column in the difference array have the same absolute values.
The main diagonal is the double of the first upper diagonal: A(n,n) = 2*A(n,n+1).
A(n,n+1) = (-1)^n*A005439(n), which also appears as the first upper diagonal of the difference array of A226158(n).

			a(0) =  0 - 2 * (-1) =  2,
a(1) = -1 - 2 * (-1) =  1,
a(2) = -1 - 2 *   0  = -1,
a(3) =  0 - 2 *   1  = -2,
a(4) =  1 - 2 *   0  =  1,
a(5) =  0 - 2 * (-3) =  6.

G. C. Greubel, Table of n, a(n) for n = 0..500
OEIS Wiki, Autosequence

Cf. A050946.

A226158 := proc(n)
    if n = 0 then
        0;
    else
        Zeta(1-n)*2*n*(2^n-1) ;
    end if;
end proc:
A230324 := proc(n)
    A226158(n)-2*A226158(n+1) ;
end proc: # R. J. Mathar, Oct 28 2013

a[0] = 2; a[1] = 1; a[n_] := n EulerE[n-1, 0] - 2 (n+1) EulerE[n, 0];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 07 2017 *)

a(n)/2 + A164555(n)/A027642(n) = 2*A225825(n)/A141056(n).

A338873 Array T(n, m) read by ascending antidiagonals: numerators of shifted Bernoulli numbers B(n, m) where m >= 0.

Original entry on oeis.org

1, -1, 1, 1, -1, 1, -1, 1, -1, 1, 1, 0, -1, -1, 1, -1, -1, 1, -19, -1, 1, 1, 0, 11, -53, -19, -1, 1, -1, 1, 43, -3113, -709, -713, -1, 1, 1, 0, -289, 349, -28813, -63367, -629, -1, 1, -1, -1, -313, 174947, -46721, -34877471, -351541, -1493, -1, 1, 1, 0, -581, 704101, -20744051, -2449743889, -176710589, -18054401, -36287, -1, 1

Offset: 0

Stefano Spezia, Nov 13 2020

			Array T(n, m):
n\m|   0       1       2       3       4 ...
---+------------------------------------
0  |   1       1       1       1       1 ...
1  |  -1      -1      -1      -1      -1 ...
2  |   1       1      -1     -19     -19 ...
3  |  -1       0       1     -53    -709 ...
4  |   1      -1      11   -3113  -28813 ...
...
Related table of shifted Bernoulli numbers B(n, m):
   1      1        1              1                1 ...
  -1   -1/2     -1/6          -1/24           -1/120 ...
   1    1/6    -1/36       -19/1440         -19/7200 ...
  -1      0    1/180      -53/11520      -709/672000 ...
   1  -1/30  11/1080  -3113/2419200  -28813/60480000 ...
  ...

Stefano Spezia, First 30 antidiagonals of the array, flattened
Takao Komatsu, Shifted Bernoulli numbers and shifted Fubini numbers, Linear and Nonlinear Analysis, Volume 6, Number 2, 2020, 245-263.

Cf. A000012 (1st row), A027641 (2nd column), A027642, A033999 (1st column), A141056, A164555, A176327, A226513 (high-order Fubini numbers), A338875, A338876.

Cf. A338874 (denominators).

B[n_,m_]:=n!Coefficient[Series[x^m/(Exp[x]-Sum[x^k/k!,{k,0,m}]+x^m),{x,0,n}],x,n]; Table[Numerator[B[n-m,m]],{n,0,10},{m,0,n}]//Flatten

T(n, m) = numerator(B(n, m)).
B(n, m) = [x^n] n!*x^m/(exp(x) - E_m(x) + x^m), where E_m(x) = Sum_{n=0..m} x^n/n! (see Equation 2.1 in Komatsu).
B(n, m) = - Sum_{k=0..n-1} n!*B(k, m)/((n - k + m)!*k!) for n > 0 (see Lemma 2.1 in Komatsu).
B(n, m) = n!*Sum_{k=1..n} (-1)^k*Sum_{i_1+...+i_k=n; i_1,...,i_k>=1} Product_{j=1..k} 1/(i_j + m)! for n > 0 (see Theorem 2.2 in Komatsu).
B(n, m) = (-1)^n*n!*det(M(n, m)) where M(n, m) is the n X n Toeplitz matrix whose first row consists in 1/(m + 1)!, 1, 0, ..., 0 and whose first column consists in 1/(m + 1)!, 1/(m + 2)!, ..., 1/(m + n)! (see Theorem 2.3 in Komatsu).
B(1, m) = -1/(m + 1)! (see Theorem 2.4 in Komatsu).
B(n, m) = n!*Sum_{t_1+2*t_2+...+n*t_n=n} (t_1,...,t_n)!*(-1)^(t_1+…+t_n)*Product_{j=1..n} (1/(m + j)!)^t_j for n >= m >= 1 (see Theorem 2.7 in Komatsu).
(-1)^n/(n + m)! = det(M(n, m)) where M(n, m) is the n X n Toeplitz matrix whose first row consists in B(1, m), 1, 0, ..., 0 and whose first column consists in B(1, m), B(2, m)/2!, ..., B(n, m)/n! (see Theorem 2.8 in Komatsu).
Sum_{k=0..n} binomial(n, k)*B(k, m)*B(n-k, m) = - n!/(m^2*m!)*Sum_{l=0..n-1} ((m! - 1)/(m*m!))^(n-l-1)*(l*(m! - 1) + m)/l!*B(l, m) - (n - m)/m*B(n, m) for m > 0 (see Theorem 4.1 in Komatsu).

A338874 Array T(n, m) read by ascending antidiagonals: denominators of shifted Bernoulli numbers B(n, m) where m >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 1, 36, 24, 1, 1, 30, 180, 1440, 120, 1, 1, 1, 1080, 11520, 7200, 720, 1, 1, 42, 9072, 2419200, 672000, 1814400, 5040, 1, 1, 1, 90720, 2322432, 60480000, 435456000, 12700800, 40320, 1, 1, 30, 38880, 232243200, 207360000, 548674560000, 21337344000, 270950400, 362880, 1

Offset: 0

Stefano Spezia, Nov 13 2020

			Array T(n, m):
n\m|   0         1         2         3         4 ...
---+--------------------------------------------
0  |   1         1         1         1         1 ...
1  |   1         2         6        24       120 ...
2  |   1         6        36      1440      7200 ...
3  |   1         1       180     11520    672000 ...
4  |   1        30      1080   2419200  60480000 ...
...
Related table of shifted Bernoulli numbers B(n, m):
   1      1        1              1                1 ...
  -1   -1/2     -1/6          -1/24           -1/120 ...
   1    1/6    -1/36       -19/1440         -19/7200 ...
  -1      0    1/180      -53/11520      -709/672000 ...
   1  -1/30  11/1080  -3113/2419200  -28813/60480000 ...
  ...

Stefano Spezia, First 30 antidiagonals of the array, flattened
Takao Komatsu, Shifted Bernoulli numbers and shifted Fubini numbers, Linear and Nonlinear Analysis, Volume 6, Number 2, 2020, 245-263.

Cf. A000012 (1st column and 1st row), A000142 (2nd row), A027641, A027642 (2nd column), A141056, A164555, A176327, A226513 (high-order Fubini numbers), A338875, A338876.

Cf. A338873 (numerators).

B[n_, m_]:=n!Coefficient[Series[x^m/(Exp[x]-Sum[x^k/k!, {k, 0, m}]+x^m), {x, 0, n}], x, n]; Table[Denominator[B[n-m,m]],{n,0,9},{m,0,n}]//Flatten

T(n, m) = denominator(B(n, m)).
B(n, m) = [x^n] n!*x^m/(exp(x) - E_m(x) + x^m), where E_m(x) = Sum_{n=0..m} x^n/n! (see Equation 2.1 in Komatsu).
B(n, m) = - Sum_{k=0..n-1} n!*B(k, m)/((n - k + m)!*k!) for n > 0 (see Lemma 2.1 in Komatsu).
B(n, m) = n!*Sum_{k=1..n} (-1)^k*Sum_{i_1+...+i_k=n; i_1,...,i_k>=1} Product_{j=1..k} 1/(i_j + m)! for n > 0 (see Theorem 2.2 in Komatsu).
B(n, m) = (-1)^n*n!*det(M(n, m)) where M(n, m) is the n X n Toeplitz matrix whose first row consists in 1/(m + 1)!, 1, 0, ..., 0 and whose first column consists in 1/(m + 1)!, 1/(m + 2)!, ..., 1/(m + n)! (see Theorem 2.3 in Komatsu).
B(1, m) = -1/(m + 1)! (see Theorem 2.4 in Komatsu).
B(n, m) = n!*Sum_{t_1+2*t_2+...+n*t_n=n} (t_1,...,t_n)!*(-1)^(t_1+…+t_n)*Product_{j=1..n} (1/(m + j)!)^t_j for n >= m >= 1 (see Theorem 2.7 in Komatsu).
(-1)^n/(n + m)! = det(M(n, m)) where M(n, m) is the n X n Toeplitz matrix whose first row consists in B(1, m), 1, 0, ..., 0 and whose first column consists in B(1, m), B(2, m)/2!, ..., B(n, m)/n! (see Theorem 2.8 in Komatsu).
Sum_{k=0..n} binomial(n, k)*B(k, m)*B(n-k, m) = - n!/(m^2*m!)*Sum_{l=0..n-1} ((m! - 1)/(m*m!))^(n-l-1)*(l*(m! - 1) + m)/l!*B(l, m) - (n - m)/m*B(n, m) for m > 0 (see Theorem 4.1 in Komatsu).

A176144 a(2n) = A164555(n). a(2n+1) = A027641(n).

Original entry on oeis.org

1, 1, 1, -1, 1, 1, 0, 0, -1, -1, 0, 0, 1, 1, 0, 0, -1, -1, 0, 0, 5, 5, 0, 0, -691, -691, 0, 0, 7, 7, 0, 0, -3617, -3617, 0, 0, 43867, 43867, 0, 0, -174611, -174611, 0, 0, 854513, 854513, 0, 0, -236364091, -236364091, 0, 0, 8553103, 8553103, 0, 0, -23749461029, -23749461029, 0, 0, 8615841276005

Offset: 0

Paul Curtz, Apr 10 2010

Formally, these are the numerators of a sequence of fractions defined by alternating A164555(n)/A027642(n) with A027641(n)/A027642(n),
which apart from the third term duplicates the Bernoulli numbers.
Essentially a duplication of the entries of A027641.

Cf. A141056, A164020, A141056.

Edited by R. J. Mathar, Jun 07 2010

A194587 A triangle whose rows add up to the numerators of the Bernoulli numbers (with B(1) = 1/2). T(n, k) for n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, -3, 4, 0, 1, -4, 3, 0, -15, 140, -270, 144, 0, 1, -20, 75, -96, 40, 0, -21, 868, -5670, 13104, -12600, 4320, 0, 1, -84, 903, -3360, 5600, -4320, 1260, 0, -15, 2540, -43470, 244944, -630000, 820800, -529200, 134400, 0, 1, -340, 9075, -74592, 278040, -544320, 582120, -322560, 72576

Offset: 0

Peter Luschny, Sep 17 2011

			[0] 1;
[1] 0,   1;
[2] 0,  -3,  4;
[3] 0,   1, -4,     3;
[4] 0, -15, 140, -270,    144;
[5] 0,   1, -20,   75,    -96,     40;
[6] 0, -21, 868, -5670, 13104, -12600,  4320;
[7] 0,   1, -84,   903, -3360,   5600, -4320, 1260;

Cf. A027641, A131689, A141056, A325871.

A194587 := proc(n, k) local i;
mul(i, i = select(isprime, map(i -> i + 1, numtheory[divisors](n)))):
(-1)^(n-k)*Stirling2(n, k) * k! / (k + 1): %%*% end:
seq(print(seq(A194587(n, k), k = 0..n)), n = 0..7);

T[n_, k_] := Times @@ Select[Divisors[n]+1, PrimeQ] (-1)^(n-k) StirlingS2[n, k]* k!/(k+1); Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Jean-François Alcover, Jun 26 2019 *)

T(n, k) = (-1)^(n - k) * A131689(n, k) * A141056(n) / (k + 1).
Sum_{k=0..n} T(n, k) = A164555(n).
T(n, n) = A325871(n).

Edited by Peter Luschny, Jun 26 2019

Edited and flipped signs in odd indexed rows by Peter Luschny, Aug 20 2022

A226157 a(n) = BS2(n) * W(n) where BS2 = sum_{k=0..n} ((-1)^k*k!/(k+1)) S_{2}(n, k) and S_{2}(n, k) are the Stirling-Frobenius subset numbers A039755(n, k). W(n) = product{p primes <= n+1 such that p divides n+1 or p-1 divides n} = A225481(n).

Original entry on oeis.org

1, 1, -2, -2, 14, 33, -62, -132, 254, 14585, -5110, -313266, 2828954, 38669001, -573370, -404801672, 237036478, 117650567067, -11499383114, -24255028327410, 1281647882998, 8203584532193105, -3584085584926, -418397193140056356, 3965530936622474, 405233976502715850633

Offset: 0

Peter Luschny, May 30 2013

a(n)/A225481(n) is case m = 2 of the scaled generalized Bernoulli numbers defined as Sum_{k=0..n} ((-1)^k*k!/(k+1)) S_{m}(n, k) where S_{m}(n, k) are Stirling-Frobenius subset numbers. A225481(n) can be seen as an analog of the Clausen numbers A141056(n).

			The numerators of 1/1, 1/2, -2/6, -2/2, 14/30, 33/6, -62/42, -132/2, 254/30, 14585/10, -5110/66, ...(the denominators are A225481(n)).

Peter Luschny, Stirling-Frobenius numbers
Peter Luschny, Generalized Bernoulli numbers.

Cf. A225481, A226156.

EulerianNumber[n_, k_, m_] := EulerianNumber[n, k, m] = If[n == 0, If[k == 0, 1 , 0], (m*(n-k) + m - 1)*EulerianNumber[n-1, k-1, m] + (m*k + 1)* EulerianNumber[n-1, k, m]];
BS[n_, m_] := Sum[Sum[EulerianNumber[n, j, m]*Binomial[j, n-k], {j, 0, n}]/ ((-m)^k*(k+1)), {k, 0, n}]
a[n_] := Product[If[Divisible[n+1, p] || Divisible[n, p-1], p, 1], {p, Prime /@ Range @ PrimePi[n+1]}] * BS[n, 2];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 27 2019, from Sage *)

@CachedFunction
def EulerianNumber(n, k, m) :   # The Eulerian numbers
    if n == 0: return 1 if k == 0 else 0
    return ((m*(n-k)+m-1)*EulerianNumber(n-1, k-1, m) +
           (m*k+1)*EulerianNumber(n-1, k, m))
@CachedFunction
def BS(n, m):   # The generalized scaled Bernoulli numbers
    return (add(add(EulerianNumber(n, j, m)*binomial(j, n - k)
           for j in (0..n))/((-m)^k*(k+1)) for k in (0..n)))
def A226157(n):   # The numerators of BS(n, 2) relative to A225481
    C = mul(filter(lambda p: ((n+1)%p == 0) or (n%(p-1) == 0), primes(n+2)))
    return C*BS(n, 2)
[A226157(n) for n in (0..25)]

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A338874 Array T(n, m) read by ascending antidiagonals: denominators of shifted Bernoulli numbers B(n, m) where m >= 0.

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