cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A165142 Numerators of a partial sum of 0, 1, 1/2, B_2, B_3, B_4,.., a modified Bernoulli sequence.

Original entry on oeis.org

0, 0, 1, 3, 5, 5, 49, 49, 58, 58, 341, 341, 1963, 1963, 14479, 14479, 39236, 39236, -2286593, -2286593, 81626353, 81626353, -928516601, -928516601, 127463912438, 127463912438, -6013599342683, -6013599342683, 149990958958943
Offset: 0

Views

Author

Paul Curtz, Sep 05 2009

Keywords

Comments

A modified list of Bernoulli numbers starts b(n) = 0, 1, 1/2, 1/6, 0, -1/30, 0, 1/42,..., n>=0, which is the standard Bernoulli sequence A027641(.)/A027642(.), prefixed with a zero and sign flipped at B_1 = -1/2.
Building partial sums of b(n) yields f(n) = 0, 0, 1, 3/2, 5/3, 5/3, 49/30, 49/30, 58/35, 58/35, 341/210, 341/210, 1963/1155,...., n>=0. The numerators of f(n) define the current sequence; denominators are found by prefixing A100650 with two 1's.
The first differences are f(n+1)-f(n) = b(n), by construction.
The inverse binomial transform of f(n) is (-1)^n*f(n); the inverse binomial transform of b(n) is 0, 1, -3/2, 5/3, -5/3, 49/30, -49/30,... an alternating sign variant of a shifted f(n).

Crossrefs

Cf. A100650 (denominators).

Programs

  • Maple
    read("transforms") ; L := [0,0,1,1/2,seq(bernoulli(i),i=2..30)] ; PSUM(L) ; apply(numer,%) ; # R. J. Mathar, Dec 02 2010
  • Mathematica
    b[n_] := BernoulliB[n-1]; b[0]=0; b[1]=1; b[2]=1/2; Join[{0}, Accumulate[ Table[b[n], {n, 0, 27}]] // Numerator] (* Jean-François Alcover_, Aug 09 2012 *)

A100649 a(n) is the numerator of 1 - Sum_{i=1..n} Bernoulli(i).

Original entry on oeis.org

1, 3, 4, 4, 41, 41, 47, 47, 289, 289, 1502, 1502, 15551, 15551, 5809, 5809, 3818123, 3818123, -76776508, -76776508, 934336415, 934336415, -127396984577, -127396984577, 6013822435553, 6013822435553, -149990624319638, -149990624319638, 167911724519886437, 167911724519886437
Offset: 0

Views

Author

N. J. A. Sloane, Dec 05 2004

Keywords

Examples

			1, 3/2, 4/3, 4/3, 41/30, 41/30, 47/35, 47/35, 289/210, 289/210, 1502/1155, 1502/1155, 15551/10010, 15551/10010, 5809/15015, 5809/15015, 3818123/510510, 3818123/510510, ...

Crossrefs

Cf. A027641, A027642, A100650 (denominators).

Programs

  • Maple
    A100649 := proc(n) 1-add( bernoulli(i),i=1..n) ; numer(%) ; end proc: # R. J. Mathar, Jul 01 2011
  • Mathematica
    a[n_]:=Numerator[1-Sum[BernoulliB[i],{i,n}]]; Array[a,30,0] (* Stefano Spezia, Aug 25 2025 *)

A176250 Numerators of the fractions defined by 2 minus partial sums of the "original" Bernoulli numbers.

Original entry on oeis.org

2, 1, 1, 1, 1, 11, 11, 12, 12, 79, 79, 347, 347, 5541, 5541, -9206, -9206, 3307613, 3307613, -78393123, -78393123, 932396477, 932396477, -127419293864, -127419293864, 6013748071263, 6013748071263
Offset: 0

Views

Author

Paul Curtz, Apr 13 2010

Keywords

Comments

We define the sequence f(n) = 2, 1, 1/2, 1/3, 1/3, 11/30, 11/30, ... for n >= 0 as 2-Sum_{i=0..n-1} A164555(i)/A027642(i). The current sequence shows the numerators of f.
Comparison with a similar sequence of fractions g(n) = A100649(n)/A100650(n): f(n) = g(n-1) - 1 for n > 1.

Crossrefs

Cf. A100650 (denominators), A164555, A027642.

Programs

  • Maple
    B := proc(n) if n = 1 then -bernoulli(n); else bernoulli(n); end if; end proc:
A176250 := proc(n) 2-add(B(i),i=0..n-1) ; numer(%) ;end proc:
seq(A176250(n),n=0..40) ; # R. J. Mathar, Jun 01 2011

A233565 Numerators of the autosequence preceding Br(n)=A229979(n)/(1 followed by A050932(n)).

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 5, 7, 7, 5, 5, 11, 11, 91, 91, -9, -9, 1207, 1207, -10849, -10849, 65879, 65879, -783127, -783127, 61098739, 61098739, -2034290233, -2034290233, 72986324461, 72986324461
Offset: 0

Views

Author

Paul Curtz, Dec 13 2013

Keywords

Comments

Br(n)=0, 1, 1, 1/2, 0, -1/6, 0, 1/6, 0, -3/10, 0, 5/6, 0, -691/210, 0,.. .
a(n) is the numerators of Bp2(n)=0, 0, 0, 1, 2, 5/2, 5/2, 7/3, 7/3, 5/2, 5/2, 11/5, 11/5, 91/30, 91/30,... . Bp2(n) is an autosequence like Br(n).
With possible future sequences we can write the array PB
1, 0, 0, 0, 0, 0, 0, 0, 0,
1, 1, 0, 0, 0, 0, 0, 0, 0,
1, 3/2, 1, 0, 0, 0, 0, 0, 0,
1, 5/3, 2, 1, 0, 0, 0, 0, 0,
1, 5/3, 5/2, 5/2, 1, 0, 0, 0, 0,
1, 49/30, 5/2, 7/2, 3, 1, 0, 0, 0,
1, 49/30, 7/3, 7/2, 14/3, 7/2, 1, 0, 0,
1, 58/35, 7/3, 3, 14/3, 6, 4, 1, 0,
1, 58/35, 5/2, 3, 7/2, 6, 15/2, 9/2, 1, etc.
The first column is A000012. The second A165142(n+1)/(1 followed by A100650(n)). The third is Bp2(n+1). The next others are built by the same way. From the second,every column is based on A164555(n)/A027642(n).
With negative (2*n+2)-th diagonals,the array without 0's is the triangle NPB. The sum of every row is
1, 0, 1/2, -1/3, 1/3, -11/30, 11/30, -12/35, 12/35, -79/210, 79/210,... .
See A176250(n+2)/A100650(n).
The inverse of NPB is A193815(n)/(A003056(n) with 1 instead of 0).

Examples

			a(0)=a(1)=0, a(i)=numerators of 0+Br(0)=0, 0+Br(1)=1, 1+Br(2)=2, 2+Br(3)=5/2, 5/2+Br(4)=5/2,... .

Crossrefs

Cf. A233316.

Programs

  • Mathematica
    nmax = 30; Br[0] = 0; Br[1] = Br[2] = 1; Br[n_] := Numerator[2*n*BernoulliB[n-1]] / Denominator[n*BernoulliB[n-1]]; Bp2 = Join[{0, 0}, Table[Br[n], {n, 0, nmax-2}] // Accumulate]; a[n_] := Numerator[Bp2[[n+1]]]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Dec 18 2013 *)

Extensions

a(17)-a(30) from Jean-François Alcover, Dec 18 2013

A172194 Numerators of the inverse binomial transform of the sequence of fractions A172030(n)/A172031(n).

Original entry on oeis.org

0, 1, 1, 2, 2, 19, 19, 23, 23, 131, 131, 808, 808, 4469, 4469, 24221, 24221, -2797103, -2797103, 80009738, 80009738, -930456539, -930456539, 127441603151, 127441603151, -6013673706973, -6013673706973, 149990847412508, 149990847412508
Offset: 0

Views

Author

Paul Curtz, Jan 29 2010

Keywords

Comments

The original sequence starts 0, 1, 5/2, 31/6, 31/3, 619/30, 619/15, 5779/70, 5779/35, 69341/210, 69341/105, ...
The inverse binomial transform yields 0, 1, 1/2, 2/3, 2/3, 19/30, 19/30, 23/35, 23/35, 131/210, 131/210, 808/1155, ... with numerators defining the sequence.
Also the numerators of the partial sums of the Bernoulli Numbers, Sum_{i=0..n} B(i). - Paul Curtz, Aug 02 2013
If we consider this sequence of partial sums b(n) := Sum_{i=0..n} B(i) = 1, 1/2, 2/3, 2/3, ... and also the sequence c(n) := 1 - Sum_{i=1..n} B(i) = 1, 3/2, 4/3, 4/3, ... mentioned in A100649, then b(n)+c(n)=2. - Paul Curtz, Aug 04 2013.

Crossrefs

Cf. A100650 (denominators), A100649, A165142.

Programs

  • Maple
    c := proc(n) option remember; if n <=1 then n; elif n = 2 then 2*procname(n-1)-bernoulli(n-1) ; else 2*procname(n-1)+bernoulli(n-1) ; end if; end proc:
L := [seq(c(n),n=0..30)] ; read("transforms") ; BINOMIALi(L) ; apply(numer,%) ; # R. J. Mathar, Dec 21 2010

A174263 Numerator of the n-th term of the inverse Binomial Transform of the Bernoulli sequence prefixed with 0.

Original entry on oeis.org

0, 1, -5, 14, -23, 349, -499, 793, -1038, 7901, -9791, 65488, -78193, 795259, -925389, 1615811, -1841036, 67142767, -75821437, 358067518, -388783203, -521129621, 480390923, 133108162049
Offset: 0

Views

Author

Paul Curtz, Mar 14 2010

Keywords

Comments

The inverse binomial transform of 0, 1, -1/2, 1/6, 0, ... is A(n) = 0, 1, -5/2, 14/3, -23/3, ... The current sequence is defined by the numerators; the denominators are A100650(n).
There is a connection to the sequence b(n) = 0, 1, 1/2, 1/6, 0, -1/30, ... of modified Bernoulli numbers [b(0)=0, b(2) = -Bernoulli(1), b(n) = Bernoulli(n-1) if n <> 2] discussed in A165142: The inverse binomial transform of b(n) is c(n) = 0, 1, -3/2, 5/3, -5/3, 49/30, -49/30, ..., and c(n) - A(n) = (-1)^n*A000217(n-1).

Crossrefs

Cf. A164558.

Programs

  • Maple
    read("transforms") ;
A174264 := proc(n) local b; b := [0,seq(bernoulli(i),i=0..n+1)] ; BINOMIALi(b) ; numer(op(n+1,%)) ; end proc:
seq(A174264(n),n=0..30) ; # R. J. Mathar, Jan 21 2011
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.