A129825 -id:A129825 - cp's OEIS frontend

A051717 1, followed by denominators of first differences of Bernoulli numbers (B(i)-B(i-1)).

1, 2, 3, 6, 30, 30, 42, 42, 30, 30, 66, 66, 2730, 2730, 6, 6, 510, 510, 798, 798, 330, 330, 138, 138, 2730, 2730, 6, 6, 870, 870, 14322, 14322, 510, 510, 6, 6, 1919190, 1919190, 6, 6, 13530, 13530, 1806, 1806, 690, 690, 282, 282, 46410, 46410, 66, 66, 1590, 1590

Offset: 0

N. J. A. Sloane

Equivalently, denominators of Bernoulli twin numbers C(n) (cf. A051716).
The Bernoulli twin numbers C(n) are defined by C(0) = 1, then C(2n) = B(2n) + B(2n-1), C(2n+1) = -B(2n+1) - B(2n), where B() are the Bernoulli numbers A027641/A027642. The definition is due to Paul Curtz.
Denominators of column 1 of table described in A051714/A051715.

			Bernoulli numbers: 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, ...
First differences: -3/2, 2/3, -1/6, -1/30, 1/30, 1/42, -1/42, -1/30, ...
Numerators: -3, 2, -1, -1, 1, 1, -1, -1, 1, 5, -5, -691, 691, 7, ...
Denominators: 2, 3, 6, 30, 30, 42, 42, 30, 30, 66, 66, 2730, ...
Sequence of C(n)'s begins: 1, -1/2, -1/3, -1/6, -1/30, 1/30, 1/42, -1/42, -1/30, 1/30, 5/66, -5/66, -691/2730, 691/2730, 7/6, -7/6, ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.

Cf. A027641, A027642, A051714, A051715, A051716, A129825, A129826.
Cf. A129724.
For numerators see A172083.

f:= func< n | Bernoulli(n) + Bernoulli(n-1) >;
function A051717(n)
  if n eq 0 then return 1;
  elif (n mod 2) eq 0 then return Denominator(f(n));
  else return Denominator(-f(n));
  end if;
end function;
[A051717(n): n in [0..50]]; // G. C. Greubel, Apr 22 2023

C:=proc(n) if n=0 then RETURN(1); fi; if n mod 2 = 0 then RETURN(bernoulli(n)+bernoulli(n-1)); else RETURN(-bernoulli(n)-bernoulli(n-1)); fi; end;

c[0]= 1; c[n_?EvenQ]:= BernoulliB[n] + BernoulliB[n-1]; c[n_?OddQ]:= -BernoulliB[n] - BernoulliB[n-1]; Table[Denominator[c[n]], {n,0,53}] (* Jean-François Alcover, Dec 19 2011 *)
Join[{1},Denominator[Total/@Partition[BernoulliB[Range[0,60]],2,1]]] (* Harvey P. Dale, Mar 09 2013 *)
Join[{1},Denominator[Differences[BernoulliB[Range[0,60]]]]] (* Harvey P. Dale, Jun 28 2021 *)

a(n)=if(n<3,n+1,denominator(bernfrac(n)-bernfrac(n-1))) \\ Charles R Greathouse IV, May 18 2015

def f(n): return bernoulli(n)+bernoulli(n-1)
def A051717(n):
    if (n==0): return 1
    elif (n%2==0): return denominator(f(n))
    else: return denominator(-f(n))
[A051717(n) for n in range(51)] # G. C. Greubel, Apr 22 2023

More terms from James Sellers, Dec 08 1999
Edited by N. J. A. Sloane, May 25 2008
Entry revised by N. J. A. Sloane, Apr 22 2021

A051716 Numerators of Bernoulli twin numbers C(n).

Original entry on oeis.org

1, -1, -1, -1, -1, 1, 1, -1, -1, 1, 5, -5, -691, 691, 7, -7, -3617, 3617, 43867, -43867, -174611, 174611, 854513, -854513, -236364091, 236364091, 8553103, -8553103, -23749461029, 23749461029, 8615841276005, -8615841276005, -7709321041217, 7709321041217, 2577687858367

Offset: 0

N. J. A. Sloane

The Bernoulli twin numbers C(n) are defined by C(0) = 1, then C(2n) = B(2n) + B(2n-1), C(2n+1) = -B(2n+1) - B(2n), where B() are the Bernoulli numbers A027641/A027642. The definition is due to Paul Curtz.
For denominators see A051717.
Negatives of numerators of column 1 of table described in A051714/A051715.

			The C(n) sequence is 1, -1/2, -1/3, -1/6, -1/30, 1/30, 1/42, -1/42, -1/30, 1/30, 5/66, -5/66, -691/2730, 691/2730, 7/6, -7/6, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..640
M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.

Cf. A000367, A027641, A027642, A051714, A051715.

Cf. A051717, A129825, A129826, A129724, A164555.

f:= func< n | Bernoulli(n) + Bernoulli(n-1) >;
function A051716(n)
  if n eq 0 then return 1;
  elif (n mod 2) eq 0 then return Numerator(f(n));
  else return Numerator(-f(n));
  end if;
end function;
[A051716(n): n in [0..50]]; // G. C. Greubel, Apr 22 2023

C:=proc(n) if n=0 then RETURN(1); fi; if n mod 2 = 0 then RETURN(bernoulli(n)+bernoulli(n-1)); else RETURN(-bernoulli(n)-bernoulli(n-1)); fi; end;

c[0]= 1; c[n_?EvenQ]:= BernoulliB[n] + BernoulliB[n-1]; c[n_?OddQ]:= -BernoulliB[n] - BernoulliB[n-1]; Table[Numerator[c[n]], {n,0,34}] (* Jean-François Alcover, Dec 19 2011 *)

a(n) = if (n==0, 1, nu = numerator(bernfrac(n)+bernfrac(n-1)); if (n%2, -nu, nu)); \\ Michel Marcus, Jan 29 2017

def f(n): return bernoulli(n)+bernoulli(n-1)
def A051716(n):
    if (n==0): return 1
    elif (n%2==0): return numerator(f(n))
    else: return numerator(-f(n))
[A051716(n) for n in range(51)] # G. C. Greubel, Apr 22 2023

Numerators of differences of the sequence of rational numbers 0 followed by A164555/A027642. - Paul Curtz, Jan 29 2017

The e.g.f. of the rationals a(n)/A051717(n) is -(1/x + x^2/2 + x/(1 - exp(x)) + dilog(exp(-x))), (with dilog(x) = polylog(2, 1-x)). From integrating the e.g.f. of the z-sequence (exp(x) - (1+x))/(exp(x) -1)^2 for the Bernoulli polynomials of the second kind (A290317 / A290318). - Wolfdieter Lang, Aug 07 2017

More terms from James Sellers, Dec 08 1999

Edited by N. J. A. Sloane, May 25 2008

A161739 The RSEG2 triangle.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 0, 13, 10, 1, 0, -4, 30, 73, 20, 1, 0, 0, -14, 425, 273, 35, 1, 0, 120, -504, 1561, 3008, 798, 56, 1, 0, 0, 736, -2856, 25809, 14572, 1974, 84, 1, 0, -12096, 44640, -73520, 125580, 218769, 55060, 4326, 120, 1

Offset: 0

Johannes W. Meijer & Nico Baken (n.h.g.baken(AT)tudelft.nl), Jun 18 2009

The EG2[2*m,n] matrix coefficients were introduced in A008955. We discovered that EG2[2m,n] = Sum_{k = 1..n} (-1)^(k+n)*t1(n-1,k-1)*2*eta(2*m-2*n+2*k)/((n-1)!)^2 with t1(n,m) the central factorial numbers A008955 and eta(m) = (1-2^(1-m))*zeta(m) with eta(m) the Dirichlet eta function and zeta(m) the Riemann zeta function.
A different way to define these matrix coefficients is EG2[2*m,n] = (1/m)*Sum_{k = 0..m-1} ZETA(2*m-2*k, n-1)*EG2[2*k, n] with ZETA(2*m, n-1) = zeta(2*m) - Sum_{k = 1..n-1} (k)^(-2*m) and EG2[0, n] = 1, for m = 0, 1, 2, ..., and n = 1, 2, 3, ... .
We define the row sums of the EG2 matrix rs(2*m,p) = Sum_{n >= 1} (n^p)*EG2(2*m,n) for p = -2, -1, 0, 1, ... and m >= p+2. We discovered that rs(2*m,p=-2) = 2*eta(2*m+2) = (1 - 2^(1-(2*m+2)))*zeta(2*m+2). This formula is quite unlike the other rs(2*m,p) formulas, see the examples.
The series expansions of the row generators RGEG2(z,2*m) about z = 0 lead to the EG2[2*m,n] coefficients while the series expansions about z = 1 lead to the ZG1[2*m-1,n] coefficients, see the formulas.
The first Maple program gives the triangle coefficients. Adding the second program to the first one gives information about the row sums rs(2*m,p).
The a(n) formulas of the right hand columns are related to sequence A036283, see also A161740 and A161741.

			The first few expressions for the ZG1[2*m-1,p+1] coefficients are:
  ZG1[2*m-1, 1] = (zeta(2*m-1))/(1/2)
  ZG1[2*m-1, 2] = (zeta(2*m-3) - zeta(2*m-1))/1
  ZG1[2*m-1, 3] = (zeta(2*m-5) - 5*zeta(2*m-3) + 4*zeta(2*m-1))/6
  ZG1[2*m-1, 4] = (zeta(2*m-7) - 14*zeta(2*m-5) + 49*zeta(2*m-3) - 36*zeta(2*m-1))/72
The first few rs(2*m,p) are (m >= p+2)
  rs(2*m, p=0) = ZG1[2*m-1,1]
  rs(2*m, p=1) = ZG1[2*m-1,1] + ZG1[2*m-1,2]
  rs(2*m, p=2) = ZG1[2*m-1,1] + 3*ZG1[2*m-1,2] + 2*ZG1[2*m-1,3]
  rs(2*m, p=3) = ZG1[2*m-1,1] + 7*ZG1[2*m-1,2] + 12*ZG1[2*m-1,3] + 6*ZG1[2*m-1,4]
The first few rs(2*m,p) are (m >= p+2)
  rs(2*m, p=-1) = zeta(2*m+1)/(1/2)
  rs(2*m, p=0) = zeta(2*m-1)/(1/2)
  rs(2*m, p=1) = (zeta(2*m-1) + zeta(2*m-3))/1
  rs(2*m, p=2) = (zeta(2*m-1) + 4*zeta(2*m-3) + zeta(2*m-5))/3
  rs(2*m, p=3) = (0*zeta(2*m-1) + 13*zeta(2*m-3) + 10*zeta(2*m-5) + zeta(2*m-7))/12
The first few rows of the RSEG2 triangle are:
  [1]
  [0, 1]
  [0, 1, 1]
  [0, 1, 4, 1]
  [0, 0, 13, 10, 1]
  [0, -4, 30, 73, 20, 1]

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 23, pp. 811-812.
J. W. Meijer and N.H.G. Baken, The Exponential Integral Distribution, Statistics and Probability Letters, Volume 5, No.3, April 1987. pp 209-211.

A000007, A129825, A161742 and A161743 are the first four left hand columns.
A000012, A000292, A107963, A161740 and A161741 are the first five right hand columns.
A010790 equals 2*r(n) and A054977 equals denom(r(n)).
A001710 equals numer(q(n)) and A141044 equals denom(q(n)).
A000142 equals the row sums.
A008955 is a central factorial number triangle.
A028246 is Worpitzky's triangle.

nmax:=10; for n from 0 to nmax do A008955(n, 0) := 1 end do: for n from 0 to nmax do A008955(n, n) := (n!)^2 end do: for n from 1 to nmax do for m from 1 to n-1 do A008955(n, m) := A008955(n-1, m-1)*n^2 + A008955(n-1, m) end do: end do: for n from 1 to nmax do A028246(n, 1) := 1 od: for n from 1 to nmax do A028246(n, n) := (n-1)! od: for n from 3 to nmax do for m from 2 to n-1 do A028246(n, m) := m*A028246(n-1, m) + (m-1)*A028246(n-1, m-1) od: od: for i from 0 to nmax-2 do s(i) := ((i+1)!/2)*sum(A028246(i+1, k1+1)*(sum((-1)^(j)*A008955(k1, j)*2*x^(2*nmax-(2*k1+1-2*j)), j=0..k1)/ (k1!*(k1+1)!)), k1=0..i) od: a(0,0) := 1: for n from 1 to nmax-1 do for m from 0 to n do a(n,m) := coeff(s(n-1), x, 2*nmax-1-2*m+2) od: od: seq(seq(a(n, m), m=0..n), n=0..nmax-1); for n from 0 to nmax-1 do seq(a(n, m), m=0..n) od;
m:=7: row := 2*m; rs(2*m, -2) := 2*eta(2*m+2); for p from -1 to m-2 do q(p+1) := (p+1)!/2 od: for p from -1 to m-2 do rs(2*m, p) := sum(a(p+1, k)*zeta(2*m+1-2*k), k=0..p+1)/q(p+1) od;

RGEG2(2*m,z) = Sum_{n >= 1} EG2[2*m,n]*z^(n-1) = Integral_{y = 0..oo}((2*y)^(2*m)/(2*m)!)* cosh(y)/(cosh(y)^2 - z)^(3/2) for m >= 0.
EG2[2*m,n] = Sum_{k = 1..n} (-1)^(k+n)* A008955(n-1, k-1)*2*eta(2*m-2*n+2*k)/((n-1)!)^2.
ZG1[2*m-1,p+1] = Sum_{j = 0..p} (-1)^j*A008955(p, j)*zeta(2*m-(2*p+1-2*j))/ r(p) with r(p)= p!*(p+1)!/2 and p >= 0.
rs(2*m,p) = Sum_{k = 0..p} A028246(p+1,k+1)*ZG1[2*m-1,k+1] and p >= 0; p <= m-2.
rs(2*m,p) = Sum_{k = 0..p+1} A161739(p+1,k)*zeta(2*m+1-2*k)/q(p+1) with q(p+1) = (p+1)!/2 and p >= -1; p <= m-2.
From Peter Bala, Mar 19 2022: (Start)
It appears that the k-th row polynomial (with indexing starting at k = 1) is given by R(k,n^2) = (k-1)!*Sum_{i = 0..n} (-1)^(n-i)*(i^k)* binomial(n,i)*binomial(n+i,i)/(n+i) for n >= 1.
For example, for k = 6, Maple's SumTools:-Summation procedure gives 5!*Sum_{i = 0..n} (-1)^(n-i)*(i^6)*binomial(n,i)*binomial(n+i,i)/(n+i) = -4*n^2 + 30*n^4 + 73*n^6 + 20*n^8 + n^10 = R(6,n^2). (End)

Minor error corrected and edited by Johannes W. Meijer, Sep 22 2012

A094310 Triangle read by rows: T(n,k), the k-th term of the n-th row, is the product of all numbers from 1 to n except k: T(n,k) = n!/k.

Original entry on oeis.org

1, 2, 1, 6, 3, 2, 24, 12, 8, 6, 120, 60, 40, 30, 24, 720, 360, 240, 180, 144, 120, 5040, 2520, 1680, 1260, 1008, 840, 720, 40320, 20160, 13440, 10080, 8064, 6720, 5760, 5040, 362880, 181440, 120960, 90720, 72576, 60480, 51840, 45360, 40320, 3628800, 1814400, 1209600, 907200, 725760, 604800, 518400, 453600, 403200, 362880

Offset: 1

Amarnath Murthy, Apr 29 2004

The sum of the rows gives A000254 (Stirling numbers of first kind). The first column and the leading diagonal are factorials given by A000142 with offsets of 0 and 1.
T(n,k) is the number of length k cycles in all permutations of {1..n}.
Second diagonal gives A001048(n). - Anton Zakharov, Oct 24 2016
T(n,k) is the number of permutations of [n] with all elements of [k] in a single cycle. To prove this result, let m denote the length of the cycle containing {1,..,k}. Letting m run from k to n, we obtain T(n,k) = Sum_{m=k..n} (C(n-k,m-k)*(m-1)!*(n-m)!) = n!/k. See an example below. - Dennis P. Walsh, May 24 2020

			Triangle begins as:
      1;
      2,     1;
      6,     3,     2;
     24,    12,     8,     6;
    120,    60,    40,    30,   24;
    720,   360,   240,   180,  144,  120;
   5040,  2520,  1680,  1260, 1008,  840,  720;
  40320, 20160, 13440, 10080, 8064, 6720, 5760, 5040;
  ...
T(4,2) counts the 12 permutations of [4] with elements 1 and 2 in the same cycle, namely, (1 2)(3 4), (1 2)(3)(4), (1 2 3)(4), (1 3 2)(4), (1 2 4)(3), (1 4 2)(3), (1 2 3 4), (1 2 4 3), (1 3 2 4), (1 3 4 2), (1 4 2 3), and (1 4 3 2). - _Dennis P. Walsh_, May 24 2020

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A000142, A000254, A001563, A001710, A002301.
Cf. A061579, A094307, A110468, A133799.
Cf. A129825. - Johannes W. Meijer, Jun 18 2009

Maple
```
seq(seq(n!/k, k=1..n), n=1..10);
```

Table[n!/k, {n,10}, {k,n}]//Flatten
Table[n!/Range[n], {n,10}]//Flatten (* Harvey P. Dale, Mar 12 2016 *)

E.g.f. for column k: x^k/(k*(1-x)).

T(n,k)*k = n*n! = A001563(n).

More terms from Philippe Deléham, Jun 11 2005

A161742 Third left hand column of the RSEG2 triangle A161739.

Original entry on oeis.org

1, 4, 13, 30, -14, -504, 736, 44640, -104544, -10644480, 33246720, 5425056000, -20843695872, -5185511654400, 23457840537600, 8506857655296000, -44092609863966720, -22430879475779174400, 130748316971139072000

Offset: 2

Johannes W. Meijer & Nico Baken (n.h.g.baken(AT)tudelft.nl), Jun 18 2009

Equals third left hand column of A161739 (RSEG2 triangle).
Other left hand columns are A129825 and A161743.
A008955 is a central factorial number triangle.
A028246 is Worpitzky's triangle.
A001710 (n!/2!), A001715 (n!/3!), A001720 (n!/4!), A001725 (n!/5!), A001730 (n!/6!), A049388 (n!/7!), A049389 (n!/8!), A049398 (n!/9!), A051431 (n!/10!) appear in Maple program.

nmax:=21; for n from 0 to nmax do A008955(n,0):=1 end do: for n from 0 to nmax do A008955(n,n):=(n!)^2 end do: for n from 1 to nmax do for m from 1 to n-1 do A008955(n,m):= A008955(n-1,m-1)*n^2+A008955(n-1,m) end do: end do: for n from 1 to nmax do A028246(n,1):=1 od: for n from 1 to nmax do A028246(n,n):=(n-1)! od: for n from 3 to nmax do for m from 2 to n-1 do A028246(n,m):=m*A028246(n-1,m)+(m-1)*A028246(n-1,m-1) od: od: for n from 2 to nmax do a(n):=sum(((-1)^k/((k+1)!*(k+2)!)) *(n!)*A028246(n,k+2)* A008955(k+1,k),k=0..n-2) od: seq(a(n),n=2..nmax);

a(n) = sum(((-1)^k/((k+1)!*(k+2)!))*(n!)*A028246(n, k+2)*A008955(k+1, k), k=0..n-2)

A161743 Fourth left hand column of the RSEG2 triangle A161739.

Original entry on oeis.org

1, 10, 73, 425, 1561, -2856, -73520, 380160, 15376416, -117209664, -7506967104, 72162155520, 7045087741056, -80246202992640, -11448278791372800, 149576169325363200, 30017051616972275712, -440857664887810867200

Offset: 3

Johannes W. Meijer & Nico Baken (n.h.g.baken(AT)tudelft.nl), Jun 18 2009

Equals fourth left hand column of A161739 (RSEG2 triangle).
Other left hand columns are A129825 and A161742.
A008955 is a central factorial number triangle.
A028246 is Worpitzky's triangle.
A001710 (n!/2!), A001715 (n!/3!), A001720 (n!/4!), A001725 (n!/5!), A001730 (n!/6!), A049388 (n!/7!), A049389 (n!/8!), A049398 (n!/9!), A051431 (n!/10!) appear in Maple program.

nmax:=21; for n from 0 to nmax do A008955(n,0):=1 end do: for n from 0 to nmax do A008955(n,n):=(n!)^2 end do: for n from 1 to nmax do for m from 1 to n-1 do A008955(n,m):= A008955(n-1,m-1)*n^2+A008955(n-1,m) end do: end do: for n from 1 to nmax do A028246(n,1):=1 od: for n from 1 to nmax do A028246(n,n):=(n-1)! od: for n from 3 to nmax do for m from 2 to n-1 do A028246(n,m):=m*A028246(n-1,m)+(m-1)*A028246(n-1,m-1) od: od: for n from 3 to nmax do a(n) := sum(((-1)^k/((k+2)!*(k+3)!))*(n!)*A028246(n,k+3)* A008955(k+2,k), k=0..n-3) od: seq(a(n),n=3..nmax);

a(n) = sum(((-1)^k/((k+2)!*(k+3)!))*(n!)*A028246(n, k+3)*A008955(k+2, k), k = 0..n-3).

A129724 a(0) = 1; then a(n) = n!(1 - (-1)^nBernoulli(n-1)).

Original entry on oeis.org

1, 2, 3, 7, 24, 116, 720, 5160, 40320, 350784, 3628800, 42940800, 479001600, 4650877440, 87178291200, 2833294464000, 20922789888000, -2166903606067200, 6402373705728000, 6808619561103360000, 2432902008176640000, -26982365129174827008000, 1124000727777607680000

Offset: 0

Paul Curtz, Jun 02 2007

sign

G. C. Greubel, Table of n, a(n) for n = 0..300

Cf. A027641, A027642, A051716, A051717, A129825, A129826.

Concatenation([1], List([1..25], n-> Factorial(n)*(1 - (-1)^n*Bernoulli(n-1)) )); # G. C. Greubel, Dec 03 2019

[n eq 0 select 1 else Factorial(n)*(1 - (-1)^n*Bernoulli(n-1)): n in [0..25]]; // G. C. Greubel, Dec 03 2019

a:= proc(n)
      if n=0 and n>=0 then 1
    elif n mod 2 = 0 then n!*(1 - bernoulli(n-1))
    else n!*(1 + bernoulli(n-1))
      fi; end;
seq(a(n), n=0..25); # modified by G. C. Greubel, Dec 03 2019

a[0] = 1; a[n_]:= n!*(1-(-1)^n*BernoulliB[n-1]); Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Sep 16 2013 *)

a(n) = if(n==0, 1, n!*(1 - (-1)^n*bernfrac(n-1)) ); \\ G. C. Greubel, Dec 03 2019

[1]+[factorial(n)*(1 - (-1)^n*bernoulli(n-1)) for n in (1..25)] # G. C. Greubel, Dec 03 2019

Edited with simpler definition by N. J. A. Sloane, May 25 2008

A137777 Triangular sequence of coefficients from the expansion of the derivative of the Bernoulli polynomial function: p(x,t) = texp(xt)/(exp(t)-1); q(x,t) = p'(x,t) = dp(x,t)/dt.

Original entry on oeis.org

2, -2, 4, 2, -12, 12, 0, 24, -72, 48, -8, 0, 240, -480, 240, 0, -240, 0, 2400, -3600, 1440, 240, 0, -5040, 0, 25200, -30240, 10080, 0, 13440, 0, -94080, 0, 282240, -282240, 80640, -24192, 0, 483840, 0, -1693440, 0, 3386880, -2903040, 725760, 0, -2177280, 0, 14515200, 0, -30481920, 0, 43545600, -32659200

Offset: 0

Roger L. Bagula and Gary W. Adamson_, Apr 28 2008

Row sums are {2, 2, 0, -8, 0, 240, 0, -24192, 0, 6048000, 0, ...}.
From Peter Luschny, Apr 23 2009: (Start)
The sequence can also be computed as the coefficients of the Bernoulli polynomials B_n(x) times 2(n+1)! for n >= 1. As Peter Pein observed the Mathematica code then reduces to
Table[CoefficientList[2 (n+1)! BernoulliB[n,x],x],{n,1,10}] // Flatten
Note that this formula is also well defined in the case n = 0 and has the value 2. (End)

			{2},
{-2, 4},
{2, -12, 12},
{0,24, -72, 48},
{-8, 0, 240, -480, 240},
{0, -240, 0, 2400, -3600, 1440},
{240, 0, -5040, 0, 25200, -30240, 10080},
{0, 13440, 0, -94080, 0, 282240, -282240, 80640},
{-24192, 0, 483840, 0, -1693440, 0, 3386880, -2903040, 725760},
{0, -2177280, 0, 14515200, 0, -30481920, 0, 43545600, -32659200, 7257600},
{6048000, 0, -119750400, 0, 399168000, 0, -558835200, 0, 598752000, -399168000, 79833600},
{0, 798336000, 0, -5269017600, 0, 10538035200, 0, -10538035200, 0, 8781696000, -5269017600, 958003200}

seq(seq(coeff(bernoulli(k,x)*2*(k+1)!,x,i),i=0..k),k=1..10); # Peter Luschny, Apr 23 2009

Clear[p, b, a]; p[t_] = D[t^2*Exp[x*t]/(Exp[t]-1),{t,1}];
a = Table[CoefficientList[2*n!^2*SeriesCoefficient
[Series[p[t],{t,0,30}],n],x],{n,0,10}]; Flatten[a]
Table[CoefficientList[2 BernoulliB[k,x] Gamma[2+k],x],{k,0,10}]//Flatten

p(x,t) = t*exp(x*t)/(exp(t)-1); q(x,t) = p'(x,t) = dp(x,t)/dt = Sum_{n>=0} Q(x,n)*t^n/n!; out_n,m=2*(n + 2)!*n!*Coefficients(Q(x,n).
A137777(n,0) = 2*A129814(n) for n >= 0.
A137777(n,n) = 2*(n+1)! for n >= 0.
Conjecture on row sums: Sum_{k=0..n+1} T(n,k) = 2*A129825(n+2). - R. J. Mathar, Jun 03 2009

Edited by N. J. A. Sloane, Jan 03 2010, incorporating comments from Peter Luschny and Peter Pein

A363041 Triangle read by rows: T(n,k) = Stirling2(n+1,k)/binomial(k+1,2) if n-k is even, else 0 (1 <= k <= n).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 5, 0, 1, 1, 0, 15, 0, 1, 0, 21, 0, 35, 0, 1, 1, 0, 161, 0, 70, 0, 1, 0, 85, 0, 777, 0, 126, 0, 1, 1, 0, 1555, 0, 2835, 0, 210, 0, 1, 0, 341, 0, 14575, 0, 8547, 0, 330, 0, 1, 1, 0, 14421, 0, 91960, 0, 22407, 0, 495, 0, 1

Offset: 1

Peter Bala, May 14 2023

A companion triangle to the triangle of Hultman numbers A164652.
The triangle of Hultman numbers can be constructed from the triangle of Stirling cycle numbers ( |A008275(n,k)| )n,k>=1 by removing the triangular number factor n*(n-1)/2 from every other entry in the n-th row (n >= 2) and setting the remaining entries to 0.
Here we carry out the analogous construction starting with the triangle of Stirling numbers of the second kind A008277, but now removing the triangular number factor k*(k+1)/2 from every other entry in the k-th column and setting the remaining entries to 0.
Do these numbers have a combinatorial interpretation?

			Triangle begins
     k = 1     2     3     4     5     6     7     8     9    10
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
n = 1:   1
    2:   0     1
    3:   1     0     1
    4:   0     5     0     1
    5:   1     0    15     0     1
    6:   0    21     0    35     0     1
    7:   1     0   161     0    70     0     1
    8:   0    85     0   777     0   126     0     1
    9:   1     0  1555     0  2835     0   210     0     1
   10:   0   341     0 14575     0  8547     0   330     0     1
   ...
Matrix product (|A008275|)^-1 * A164652 * A008277 begins
  / 1             \ /1         \ /1           \   /1           \
  |-1   1         | |0 1       | |1  1        |   |0 1         |
  | 1  -3  1      | |1 0  1    | |1  3  1     | = |0 0 1       |
  |-1   7 -6   1  | |0 5  0 1  | |1  7  6  1  |   |0 1 0  1    |
  | 1 -15 25 -10 1| |8 0 15 0 1| |1 15 25 10 1|   |0 0 5  0 1  |
  | ...           | |...       | |...         |   |0 1 0 15 0 1|
  |               | |          | |            |   |...         |

Row sums give A363042.

Cf. A008275, A008277, A164652, A129825.

A362041:= (n, k)-> `if`(n-k mod 2 = 0, Stirling2(n+1,k)/binomial(k+1,2), 0):
for n from 1 to 10 do seq(A362041(n,k), k = 1..n) od;

T(n,k) = if ((n-k) % 2, 0, stirling(n+1, k, 2)/binomial(k+1, 2)); \\ Michel Marcus, May 23 2023

Let P(n,x) = (1 - x)*(1 - 2*x)*...*(1 - n*x). The g.f. for the k-th column of the triangle is (1/(k*(k + 1)))*x^(k-1)*(1/P(k,x) - 1/P(k,-x)) = (x^k)*(x^k*R(k-1,1/x))/((1 - x^2)*(1 - 4*x^2)*...*(1 - k^2*x^2)), where R(n,x) denotes the n-th row polynomial of A164652. (Since the entries of triangle A164652 are integers, it follows that the entries of the present triangle are also integers.)
It appears that the matrix product (|A008275|)^-1 * A164652 * A008277 =  I_1 + A363041 (direct sum, where I_1 is the 1 X 1 identity matrix). See the Example section.
The sequence of row sums of the inverse array begins [1, 1, 0, -4, 0, 120, 0, -12096, 0, 3024000, 0, -1576143360, 0, 1525620096000, 0, -2522591034163200, 0, 6686974460694528000, 0, -27033456071346536448000, ...], and appears to be essentially A129825.

cp's OEIS Frontend

A051717 1, followed by denominators of first differences of Bernoulli numbers (B(i)-B(i-1)).

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A051716 Numerators of Bernoulli twin numbers C(n).

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A161739 The RSEG2 triangle.

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A094310 Triangle read by rows: T(n,k), the k-th term of the n-th row, is the product of all numbers from 1 to n except k: T(n,k) = n!/k.

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A161742 Third left hand column of the RSEG2 triangle A161739.

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A161743 Fourth left hand column of the RSEG2 triangle A161739.

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A129724 a(0) = 1; then a(n) = n!*(1 - (-1)^n*Bernoulli(n-1)).

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A129724 a(0) = 1; then a(n) = n!(1 - (-1)^nBernoulli(n-1)).

A137777 Triangular sequence of coefficients from the expansion of the derivative of the Bernoulli polynomial function: p(x,t) = texp(xt)/(exp(t)-1); q(x,t) = p'(x,t) = dp(x,t)/dt.