A051715 -id:A051715 - 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

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

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.

Original entry on oeis.org

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

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

A085738 Denominators in triangle formed from Bernoulli numbers.

Original entry on oeis.org

1, 2, 2, 6, 3, 6, 1, 6, 6, 1, 30, 30, 15, 30, 30, 1, 30, 15, 15, 30, 1, 42, 42, 105, 105, 105, 42, 42, 1, 42, 21, 105, 105, 21, 42, 1, 30, 30, 105, 105, 105, 105, 105, 30, 30, 1, 30, 15, 105, 105, 105, 105, 15, 30, 1, 66, 66, 165, 165, 1155, 231, 1155, 165, 165, 66, 66

Offset: 0

N. J. A. Sloane following a suggestion of J. H. Conway, Jul 23 2003

Triangle is determined by rules 0) the top number is 1; 1) each number is the sum of the two below it; 2) it is left-right symmetric; 3) the numbers in each of the border rows, after the first 3, are alternately 0.

Up to signs this is the difference table of the Bernoulli numbers (see A212196). The Sage script below is based on L. Seidel's algorithm and does not make use of a library function for the Bernoulli numbers; in fact it generates the Bernoulli numbers on the fly. - Peter Luschny, May 04 2012

			Triangle begins
    1
   1/2,   1/2
   1/6,   1/3,   1/6
    0,    1/6,   1/6,     0
  -1/30,  1/30,  2/15,   1/30,  -1/30
    0,   -1/30,  1/15,   1/15,  -1/30,     0
   1/42, -1/42, -1/105,  8/105, -1/105,  -1/42,   1/42
    0,    1/42, -1/21,   4/105,  4/105,  -1/21,   1/42,   0
  -1/30,  1/30, -1/105, -4/105,  8/105,  -4/105, -1/105, 1/30, -1/30

Fabien Lange and Michel Grabisch, The interaction transform for functions on lattices Discrete Math. 309 (2009), no. 12, 4037-4048. [From _N. J. A. Sloane_, Nov 26 2011]
Peter Luschny, The computation and asymptotics of the Bernoulli numbers.
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [_Peter Luschny_, May 04 2012]

See A051714/A051715 for another triangle that generates the Bernoulli numbers.

Cf. A085737, A212196.

t[n_, 0] := (-1)^n BernoulliB[n];
t[n_, k_] := t[n, k] = t[n-1, k-1] - t[n, k-1];
Table[t[n, k] // Denominator, {n, 0, 10}, {k, 0, n}] (* Jean-François Alcover, Jun 04 2019 *)

# uses[BernoulliDifferenceTable from A085737]
def A085738_list(n): return [q.denominator() for q in BernoulliDifferenceTable(n)]
A085738_list(6)
# Peter Luschny, May 04 2012

T(n, 0) = (-1)^n*Bernoulli(n); T(n, k) = T(n-1, k-1) - T(n, k-1) for k=1..n. [Corrected (sign flipped) by R. J. Mathar, Jun 02 2010]

Let U(m, n) = (-1)^(m + n)*T(m+n, n). Then the e.g.f. for U(m, n) is (x - y)/(e^x - e^y). - Ira M. Gessel, Jun 12 2021

cp's OEIS Frontend

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

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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.

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