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 1-3 of 3 results.

A100650 Denominator of 1 - Sum_{i=1..n} Bernoulli(i).

Original entry on oeis.org

1, 2, 3, 3, 30, 30, 35, 35, 210, 210, 1155, 1155, 10010, 10010, 15015, 15015, 510510, 510510, 1616615, 1616615, 1939938, 1939938, 22309287, 22309287, 74364290, 74364290, 111546435, 111546435, 6469693230, 6469693230, 33426748355, 33426748355, 200560490130, 200560490130
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, ...

Links

Crossrefs

Cf. A002110, A100649 (numerators).

Programs

  • Maple
    A100650 := proc(n) 1-add( bernoulli(i),i=1..n) ; denom(%) ; end proc: # R. J. Mathar, Jul 01 2011
  • Mathematica
    Denominator[1-Accumulate[BernoulliB[Range[0,40]]]] (* Harvey P. Dale, Feb 19 2015 *)
  • PARI
    a(n) = denominator(1 - sum(i=1, n, bernfrac(i))); \\ Michel Marcus, Feb 15 2021

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

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
Showing 1-3 of 3 results.

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

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