A051714 -id:A051714 - cp's OEIS frontend

A051718 Numerators of column 2 of table described in A051714/A051715.

1, 1, 3, 1, -3, -1, 1, 1, 1, -5, -1017, 691, 601, -7, -809, 3617, 922191, -43867, -6132631, 174611, 12988703, -854513, -1552922421, 236364091, 1139644561, -8553103, -7089687053, 23749461029, 378639019356093, -8615841276005

Offset: 0

N. J. A. Sloane

Seiichi Manyama, Table of n, a(n) for n = 0..627
M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.

Cf. A051714, A051715, A051719.

nmax = 29; 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]] // Numerator (* Jean-François Alcover, Oct 08 2012 *)

a(n) = numerator(n! * [x^n] f(x)) where f(x) = -(x*exp(3*x))/(1-exp(x))^3+5/(2*(1-exp(x)))-1/(1-exp(x))^2-5/6. - Vladimir Kruchinin, Nov 03 2015

More terms from James Sellers, Dec 08 1999

A051720 Numerators of column 3 of table described in A051714/A051715.

Original entry on oeis.org

1, 1, 2, 2, -1, -4, -1, 8, 7, -44, -2663, 368, 1247, -244, -1511, 43416, 1623817, -276356, -10405289, -21376, 21491081, 32209348, -2523785339, -107638072, 1827648887, 842271812, -11254630547, -17380760743952, 596303510772251

Offset: 0

N. J. A. Sloane

Seiichi Manyama, Table of n, a(n) for n = 0..626
M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.

Cf. A051721.

a[0, k_] := 1/(k+1); a[n_, k_] := a[n, k] = (k+1)*(a[n-1, k] - a[n-1, k+1]); a[n_] := a[n, 3] // Numerator; Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Sep 17 2012 *)

a(n) = numerator(n! * [x^n] f(x)) where f(x) =(x*exp(4*x))/(1-exp(x))^4+13/(3*(1-exp(x)))-7/(2*(1-exp(x))^2)+1/(1-exp(x))^3-13/12. - Vladimir Kruchinin, Nov 03 2015

More terms from James Sellers, Dec 08 1999

A051723 Denominators of row 4 of table described in A051714/A051715.

Original entry on oeis.org

30, 30, 140, 105, 1, 140, 3960, 495, 1430, 6006, 5460, 130, 7140, 2040, 5168, 14535, 11970, 14630, 15180, 5313, 6325, 89700, 23400, 6825, 142506, 7830, 125860, 53940, 40920, 92752, 628320, 6545, 6290, 442890, 329004, 45695, 151905, 223860, 493640

Offset: 0

N. J. A. Sloane

			-1/30 -1/30 -3/140 -1/105 0 ...

M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.

Cf. A051722.

More terms from James Sellers, Dec 08 1999

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

A051721 Denominators of column 3 of table described in A051714/A051715.

Original entry on oeis.org

4, 5, 15, 35, 105, 105, 105, 165, 165, 455, 15015, 1365, 1365, 255, 255, 11305, 33915, 21945, 21945, 345, 3795, 10465, 31395, 1365, 1365, 435, 435, 346115, 1038345, 55335, 608685, 255, 255, 319865, 959595, 959595, 959595, 6765, 6765, 61705

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. A051720.

a[0, k_] := 1/(k+1); a[n_, k_] := a[n, k] = (k+1)*(a[n-1, k] - a[n-1, k+1]); a[n_] := a[n, 3] // Denominator; Table[a[n], {n, 0, 39}] (* Jean-François Alcover, Sep 17 2012 *)

More terms from James Sellers, Dec 08 1999

A051722 Numerators of row 4 of table described in A051714/A051715.

Original entry on oeis.org

-1, -1, -3, -1, 0, 1, 49, 8, 27, 125, 121, 3, 169, 49, 125, 352, 289, 351, 361, 125, 147, 2057, 529, 152, 3125, 169, 2673, 1127, 841, 1875, 12493, 128, 121, 8381, 6125, 837, 2738, 3971, 8619, 1000, 1681, 1813, 35131, 1573, 3375, 21689, 2209, 4128, 26411

Offset: 0

N. J. A. Sloane

			-1/30 -1/30 -3/140 -1/105 0 ...

M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.

Cf. A051723.

More terms from James Sellers, Dec 08 1999

A194531 Numerator of row 4 in A051714(n) or row 3 in A176672(n).

Original entry on oeis.org

0, 1, 1, 2, 5, 5, 7, 28, 3, 15, 55, 22, 13, 91, 35, 40, 34, 51, 57, 190, 35, 77, 253, 92, 25, 325, 117, 126, 203, 145, 155, 496, 44, 187, 595, 210, 111, 703, 247, 260, 205, 287, 301, 946, 165, 345, 1081, 376, 98, 1225, 425

Offset: 0

Paul Curtz, Aug 28 2011

Akiyama-Tanigawa algorithm from 1/n leads to Bernoulli A164555(n)/A027642(n):
1,   1/2,  1/3,  1/4,
1/2, 1/3,  1/4,  1/5,
1/6, 1/6, 3/20, 2/15,           =A026741(n+1)/A045896(n+1),
0,  1/30, 1/20, 2/35, 5/84, 5/84, 7/120, 28/495  =a(n)/b(n).

Cf. A193220 (denominators).

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[3, k], {k, 0, 50}] // Numerator (* Jean-François Alcover, Sep 19 2012 *)

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

Original entry on oeis.org

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

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

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A051718 Numerators of column 2 of table described in A051714/A051715.

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A051720 Numerators of column 3 of table described in A051714/A051715.

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A051723 Denominators of row 4 of table described in A051714/A051715.

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

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A051721 Denominators of column 3 of table described in A051714/A051715.

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A051722 Numerators of row 4 of table described in A051714/A051715.

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A194531 Numerator of row 4 in A051714(n) or row 3 in A176672(n).

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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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