A051718 -id:A051718 - cp's OEIS frontend

A051714 Numerators of table a(n,k) read by antidiagonals: a(0,k) = 1/(k+1), a(n+1,k) = (k+1)*(a(n,k) - a(n,k+1)), n >= 0, k >= 0.

1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 3, 1, -1, 1, 1, 2, 1, -1, 0, 1, 1, 5, 2, -3, -1, 1, 1, 1, 3, 5, -1, -1, 1, 0, 1, 1, 7, 5, 0, -4, 1, 1, -1, 1, 1, 4, 7, 1, -1, -1, 1, -1, 0, 1, 1, 9, 28, 49, -29, -5, 8, 1, -5, 5, 1, 1, 5, 3, 8, -7, -9, 5, 7, -5, 5, 0, 1, 1, 11, 15, 27, -28, -343, 295, 200, -44, -1017, 691, -691

Offset: 0

N. J. A. Sloane

Leading column gives the Bernoulli numbers A164555/A027642. - corrected by Paul Curtz, Apr 17 2014

			Table begins:
   1     1/2   1/3    1/4   1/5  1/6  1/7 ...
   1/2   1/3   1/4    1/5   1/6  1/7 ...
   1/6   1/6   3/20   2/15  5/42 ...
   0     1/30  1/20   2/35  5/84 ...
  -1/30 -1/30 -3/140 -1/105 ...
Antidiagonals of numerator(a(n,k)):
  1;
  1,  1;
  1,  1,  1;
  1,  1,  1,  0;
  1,  1,  3,  1, -1;
  1,  1,  2,  1, -1,   0;
  1,  1,  5,  2, -3,  -1,  1;
  1,  1,  3,  5, -1,  -1,  1,  0;
  1,  1,  7,  5,  0,  -4,  1,  1, -1;
  1,  1,  4,  7,  1,  -1, -1,  1, -1,  0;
  1,  1,  9, 28, 49, -29, -5,  8,  1, -5,  5;

Alois P. Heinz, Antidiagonals n = 0..140, flattened
M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.
Index entries for sequences related to Bernoulli numbers.

Rows 2, 3, 4 give: A026741/A045896, A051712/A051713, A051722/A051723.
Columns 0, 1, 2, 3 give: A000367/A002445, A051716/A051717, A051718/A051719, A051720/A051721.
Denominators are in A051715.
Cf. A027642, A164555.

function a(n,k)
  if n eq 0 then return 1/(k+1);
  else return (k+1)*(a(n-1,k) - a(n-1,k+1));
  end if;
end function;
A051714:= func< n,k | Numerator(a(n,k)) >;
[A051714(k,n-k): k in [0..n], n in [0..15]]; // G. C. Greubel, Apr 22 2023

a:= proc(n,k) option remember;
      `if`(n=0, 1/(k+1), (k+1)*(a(n-1,k)-a(n-1,k+1)))
    end:
seq(seq(numer(a(n, d-n)), n=0..d), d=0..12); # Alois P. Heinz, Apr 17 2013

nmax = 12; a[0, k_]:= 1/(k+1); a[n_, k_]:= a[n, k]= (k+1)(a[n-1, k]-a[n-1, k+1]); Numerator[Flatten[Table[a[n-k, k], {n,0,nmax}, {k, n, 0, -1}]]] (* Jean-François Alcover, Nov 28 2011 *)

def a(n,k):
    if (n==0): return 1/(k+1)
    else: return (k+1)*(a(n-1, k) - a(n-1, k+1))
def A051714(n,k): return numerator(a(n, k))
flatten([[A051714(k, n-k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Apr 22 2023

From Fabián Pereyra, Jan 14 2023: (Start)
a(n,k) = numerator(Sum_{j=0..n} (-1)^(n-j)*j!*Stirling2(n,j)/(j+k+1)).
E.g.f.: A(x,t) = (x+log(1-t))/(1-t-exp(-x)) = (1+(1/2)*x+(1/6)*x^2/2!-(1/30)*x^4/4!+...)*1 + (1/2+(1/3)*x+(1/6)*x^2/2!+...)*t + (1/3+(1/4)*x+(3/20)*x^2/2!+...)*t^2 + .... (End)

More terms from James Sellers, Dec 07 1999

A051715 Denominators of table a(n,k) read by antidiagonals: a(0,k) = 1/(k+1), a(n+1,k) = (k+1)(a(n,k)-a(n,k+1)), n >= 0, k >= 0.

Original entry on oeis.org

1, 2, 2, 3, 3, 6, 4, 4, 6, 1, 5, 5, 20, 30, 30, 6, 6, 15, 20, 30, 1, 7, 7, 42, 35, 140, 42, 42, 8, 8, 28, 84, 105, 28, 42, 1, 9, 9, 72, 84, 1, 105, 140, 30, 30, 10, 10, 45, 120, 140, 28, 105, 20, 30, 1, 11, 11, 110, 495, 3960, 924, 231, 165, 220, 66, 66, 12, 12, 66, 55, 495, 264, 308, 132, 165, 44, 66, 1

Offset: 0

N. J. A. Sloane

Leading column gives the Bernoulli numbers A027641/A027642.

			Table begins:
    1    1/2   1/3    1/4   1/5  1/6  1/7 ...
   1/2   1/3   1/4    1/5   1/6  1/7 ...
   1/6   1/6   3/20   2/15  5/42 ...
    0    1/30  1/20   2/35  5/84 ...
  -1/30 -1/30 -3/140 -1/105 ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.
Index entries for sequences related to Bernoulli numbers.

Rows 2, 3, 4 give A026741/A045896, A051712/A051713, A051722/A051723.
Columns 0, 1, 2, 3 give A000367/A002445, A051716/A051717, A051718/A051719, A051720/A051721.
Numerators are in A051714.

a:= proc(n,k) option remember;
      `if`(n=0, 1/(k+1), (k+1)*(a(n-1,k)-a(n-1,k+1)))
    end:
seq(seq(denom(a(n, d-n)), n=0..d), d=0..12); # Alois P. Heinz, Apr 17 2013

nmax = 12; a[0, k_] := 1/(k+1); a[n_, k_] := a[n, k] = (k+1)(a[n-1, k]-a[n-1, k+1]); Denominator[ Flatten[ Table[ a[n-k, k], {n, 0, nmax}, {k, n, 0, -1}]]](* Jean-François Alcover, Nov 28 2011 *)

a(n,k) = denominator(Sum_{j=0..n} (-1)^(n-j)*j!*Stirling2(n,j)/(j+k+1)). - Fabián Pereyra, Jan 14 2023

More terms from James Sellers, Dec 08 1999

A129826 Transformed Bernoulli twin numbers.

Original entry on oeis.org

1, -1, -2, -4, -4, 24, 120, -960, -12096, 120960, 3024000, -36288000, -1576143360, 22066007040, 1525620096000, -24409921536000, -2522591034163200, 45406638614937600, 6686974460694528000, -133739489213890560000, -27033456071346536448000, 594736033569623801856000

Offset: 0

Paul Curtz, May 20 2007

sign

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

Cf. A027641, A027642, A051717, A051718, A129378, A140351.

f:= func< n | n le 2 select (-1)^Floor((n+1)/2)/(n+1) else (-1)^n*BernoulliNumber(Floor(n - (1-(-1)^n)/2)) >;
A129826:= func< n | Factorial(n+1)*f(n) >;
[A129826(n): n in [0..30]]; // G. C. Greubel, Feb 01 2024

c[n_?EvenQ] := BernoulliB[n]; c[n_?OddQ] := -BernoulliB[n-1]; c[1]=-1/2; c[2]=-1/3; a[n_] := (n+1)!*c[n]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Aug 08 2012 *)

def f(n): return (-1)^((n+1)//2)/(n+1) if n<3 else (-1)^n*bernoulli(n-(n%2))
def A129826(n): return factorial(n+1)*f(n)
[A129826(n) for n in range(31)] # G. C. Greubel, Feb 01 2024

We define Bernoulli twin numbers C(n) via Bernoulli numbers B(n) = A027641(n)/A027642(n) as C(0)=1, 2C(1)=-1, 3C(2)=-1, C(2n-1)= -B(2n-2) and C(2n)=B(2n), n>1. The sequence is defined as a(n)=(n+1)!*C(n).
a(n) = (n+1)!*C(n), where C(n) = A051718(n)/A051717(n).
E.g.f.: Sum(n>=0) C(n) x^n/n! = 1 + x - x^2/2 + Sum_{n>=1} (B(n) - B(n-1))*x^n/n! = x - x^2/2 + x/(e^x-1) - Integral_{y=0..x} ((y dy)/(e^y-1)).

Edited and extended by R. J. Mathar, Aug 06 2008

A051719 Denominators of column 2 of table described in A051714/A051715.

Original entry on oeis.org

3, 4, 20, 20, 140, 28, 140, 20, 220, 44, 20020, 1820, 1820, 4, 340, 340, 45220, 532, 29260, 220, 5060, 92, 41860, 1820, 1820, 4, 580, 580, 1384460, 9548, 811580, 340, 340, 4, 1279460, 1279460, 1279460, 4, 9020, 9020, 2715020, 1204, 138460, 460

Offset: 0

N. J. A. Sloane

Seiichi Manyama, Table of n, a(n) for n = 0..1000
M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.

Cf. A051714, A051715, A051718.

nmax = 43; a[0, k_] := 1/(k + 1); a[n_, k_] := a[n, k] = (k + 1)*(a[n - 1, k] - a[n - 1, k + 1]); Table[a[n, k], {n, 0, nmax}, {k, 0, nmax}] [[All, 3]] // Denominator (* Jean-François Alcover, Oct 08 2012 *)

More terms from James Sellers, Dec 08 1999

cp's OEIS Frontend

A051714 Numerators of table a(n,k) read by antidiagonals: a(0,k) = 1/(k+1), a(n+1,k) = (k+1)*(a(n,k) - a(n,k+1)), n >= 0, k >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

Extensions

A051715 Denominators of table a(n,k) read by antidiagonals: a(0,k) = 1/(k+1), a(n+1,k) = (k+1)(a(n,k)-a(n,k+1)), n >= 0, k >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A129826 Transformed Bernoulli twin numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

A051719 Denominators of column 2 of table described in A051714/A051715.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions