A172083 -id:A172083 - cp's OEIS frontend

A172086 Numerators of sum (C(n) = A051716/A051717) + (1 followed by first differences A172083/A051717 of Bernoulli numbers).

2, -2, 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, -2577687858367

Offset: 0

Paul Curtz, Jan 25 2010

Denominators: 1, 1, 3, 3, 15, 15, 21, 21, 15, 15, 33, 33, 1365, 1365, ... = A001897 with terms repeated. See A000367/A002445.

Cf. A000367, A001897, A002445, A027641, A051716, A051717

Name edited by Michel Marcus, Jan 30 2021

Clarified definition, added more terms. - N. J. A. Sloane, Apr 22 2021

A176511 From Bernoulli twin numbers to Catalan numbers arrays (*).First part.We consider array, from Bernoulli twin numbers A051716/A051717 mixed with their companion A172083/A051717 BTC(n)=1,1,-1/2,-3/2,-1/3,2/3,-1/6,-1/6, and successive differences ,named BTC1. a(n) are numerators of BTC(n).Denominators are (double A051717)=1,1,2,2,3,3,6,6,30,30,30,30,.

Original entry on oeis.org

1, 1, -1, -3, -1, 2, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 5, 5

Offset: 0

Paul Curtz, Apr 19 2010

(*) Even case:ECT(n) in A176239. BTC(n) are not Bi-Bernoulli numbers (absolute values of mixed sequences are not the same like BB1(n) in A176144 or BB2(n) in A176184). Rows of array BTC1: 1) 1,1,-1/2,-3/2,-1/3,2/3,-1/6,-1/6,-1/30,-1/30,1/30,1/30,1/42,1/42,; 2) 0,-3/2,-1,7/6,1,-5/6,0,2/15,0,2/15,0; 3) -3/2,1/2,13/6,-1/6,-11/6,5/6,2/15,-2/15,2/15,-2/15; 4) 2,5/3,-7/3,-5/3,8/3,-7/10,-4/15,4/15,-4/15; 5) -1/3,-4,2/3,13/3,-101/30,13/30,8/15,-8/15; 6) -11/3,14/3,11/3,-77/10,19/5,1/10,-16/15; 7) 25/3,-1, -341/30,23/2,-37/10,-29/30; 8) -28/3,-311/30,343/15,. Correction:in A176150 last term (-517) is false.

A176784 From Bernoulli twin numbers to Catalan numbers arrays.Second part.Consider array from companion of Bernoulli twin numbers A172083/A051717 mixed with A051716/A051717 BCT(n)=1,1,-3/2,-1/2,2/3,-1/3,-1/6,-1/6, with successive differences,named BCT1.a(n) are numerators of BCT(n).Denominators (double A051717)=1,1,2,2,3,3,6,6,30,30,30,30,.

Original entry on oeis.org

1, 1, -3, -1, 2, -1, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 5, 5

Offset: 0

Paul Curtz, Apr 26 2010

a(n) is A176511 (companion) with A176511(2),A176511(3), A176511(4),A176511(5) swapped by pairs.Rows of BCT1: 1) 1,1,-3/2,-1/2,2/3,-1/3,-1/6,-1/6; 2) 0,-5/2,1,7/6,-1,1/6,0,2/15; 3) -5/2,7/2,1/6,-13/6,7/6,-1/6,2/15,-2/15; 4) 6,-10/3,-7/3,10/3,-4/3,3/10,-4/15,4/15; 5) -28/3,1,17/3,-14/3,49/30,-17/30,8/15,-8/15; 6) 31/3,14/3,-31/3,63/10,-11/5,11/10,16/15; 7) -17/3,-15,499/30,-17/2,33/10,-1/30; 8) -28/3,949/30,-377/15; .Now we subtract first part BTC1 and second BCT1.Hence an array with only integers.We consider it from seventh column from right to left.Columns changed into rows give different possibilities for Catalan numbers A000108 or A000108(n+1). Among them,ECT(n) in A176239. Odd triangle is 1, 1,0,-1, 0,1,-1,0,2, 0,0,1,-2,2,0,-5, 0,0,0,1,-3,5,-5,0,14, .

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

cp's OEIS Frontend

A172086 Numerators of sum (C(n) = A051716/A051717) + (1 followed by first differences A172083/A051717 of Bernoulli numbers).

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

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