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.

A195240 Numerators of the second differences of the sequence of fractions (-1)^(n+1)*A176618(n)/A172031(n).

Original entry on oeis.org

0, 1, 1, 7, 8, 11, 10, 7, 8, 19, 14, 337, 1028, 5, -2, -1681, 1936, 22133, -21734, -87223, 87388, 427291, -427222, -118181363, 118182728, 4276553, -4276550, -11874730297, 11874730732, 4307920641583
Offset: 0

Views

Author

Paul Curtz, Sep 13 2011

Keywords

Comments

The array of (-1)^n*A176328(n)/A176591(n) and its first, second, etc. differences in subsequence rows starts as follows:
0, 1, 2, 19/6, 14/3, 199/30, 137/15, ... (-1)^n * A176328(n)/A176591(n),
1, 1, 7/6, 3/2, 59/30, 5/2, 127/42, ... see A176328,
0, 1/6, 1/3, 7/15, 8/15, 11/21, 10/21, ...
1/6, 1/6, 2/15, 1/15, -1/105, -1/21, -1/105, ... see A190339
0, -1/30, -1/15, -8/105, -4/105, 4/105, -116/1155, ...
The numerators in the 3rd row, 0, 1/6, 1/3, 7/15, 8/15, 11/21, 10/21, 7/15, 8/15, 19/33, 14/33, 337/1365, 1028/1365, 5/3, -2/3, -1681/255, 1936/255, ... define the current sequence.
The associated denominators are 1, 6 and followed by 3, 15, 15 etc as provided in A172087.
The second column of the array, 1, 1, 1/6, 1/6, -1/30, -1/30, ... contains doubled A000367(n)/A002445(n). These are related to A176150, A176144, and A176184.
In the first subdiagonal of the array we see 1, 1/6, 2/15, -8/150, 8/105, -32/321, 6112/15015, -3712/2145 , ... continued as given by A181130 and A181131.

Links

Programs

  • Maple
    read("transforms") ;
evb := [0, 1, 0, seq(bernoulli(n), n=2..30)] ;
ievb := BINOMIALi(evb) ;
[seq((-1)^n*op(n,ievb),n=1..nops(ievb))] ;
DIFF(%) ;
DIFF(%) ;
apply(numer,%) ; # R. J. Mathar, Sep 20 2011
  • Mathematica
    evb = Join[{0, 1, 0}, Table[BernoulliB[n], {n, 2, 32}]]; ievb = Table[ Sum[Binomial[n, k]*evb[[k+1]], {k, 0, n}], {n, 0, Length[evb]-3}]; Differences[ievb, 2] // Numerator (* Jean-François Alcover, Sep 09 2013, after R. J. Mathar *)

Formula

a(2*n+1) + a(2*n+2) = A172087(2*n+2) = A172087(2*n+3), n >= 1.

A256675 Denominators of the inverse binomial transform of Bernoulli(n+2).

Original entry on oeis.org

6, 6, 15, 15, 105, 21, 105, 15, 165, 33, 15015, 1365, 1365, 3, 255, 255, 33915, 399, 21945, 165, 3795, 69, 31395, 1365, 1365, 3, 435, 435, 1038345, 7161, 608685, 255, 255, 3, 959595, 959595, 959595, 3, 6765, 6765, 2036265, 903, 103845, 345, 16215, 141, 1090635
Offset: 0

Views

Author

Paul Curtz, Apr 07 2015

Keywords

Comments

Difference table of B(n+2):
1/6, 0, -1/30, 0, 1/42, 0, -1/30, ...
-1/6, -1/30, 1/30, 1/42, -1/42, -1/30, ...
2/15, 1/15, -1/105, -1/21, -1/105, ...
-1/15, -8/105, -4/105, 4/105, ...
-1/105, 4/105, 8/105, ...
1/21, 4/105, ...
-1/105, ...
...
a(n) is the denominator of the n-th term of the first column.
a(n+2) is the denominator of the n-th term of the third row.
See A239315(n), which is the table without the first two rows.
Inverse binomial transform: 1/6, -1/6, 2/15, -1/15, -1/105, 1/21, -1/105, -1/15, 7/165, 5/33, -2663/15015, ... .

Crossrefs

Cf. A190339, A239315, A029765, A001897, A141459, A172087, A168516, A177735, A256595.

Programs

  • Mathematica
    max = 42; bb = Table[BernoulliB[n+2], {n, 0, max}]; dd = Table[Differences[bb, n], {n, 0, max}]; dd[[All, 1]] // Denominator (* Jean-François Alcover, Apr 09 2015 *)
  • PARI
    lista(nn) = {A = vector(nn, n, bernfrac(n+1)); for (i=1, #A-1, for(j=0,i-1,A[i+1]-=binomial(i,j)*A[j+1])); for (i=1, #A, print1(denominator(A[i]), ", "));} \\ Michel Marcus, Apr 08 2015

Formula

a(2n) = A029765(n).
a(2n+3) = A001897(n+2).
a(2n)/a(2n+1) = A177735(n).
a(2n+4)/a(2n+3) = A177735(n+3).

A256003 a(n) = 0 followed by numerators of 2*A176327(n)/A176289(n).

Original entry on oeis.org

0, 2, 0, 1, 0, -1, 0, 1, 0, -1, 0, 5, 0, -691, 0, 7, 0, -3617, 0, 43867, 0, -174611, 0, 854513, 0, -236364091, 0, 8553103, 0, -23749461029, 0, 8615841276005, 0, -7709321041217, 0, 2577687858367, 0, -26315271553053477373, 0
Offset: 0

Views

Author

Paul Curtz, May 06 2015

Keywords

Comments

Offset 0 is chosen instead of -1. (The offset 0 corresponds to A176327(n), -1 to 0 followed by A176327(n).)
Denominators: b(n) = 1 followed by A141459(n).
Difference table of a(n)/b(n):
0, 2, 0, 1/3, 0, -1/15, 0, ...
2, -2, 1/3, -1/3, -1/15, 1/15, ...
-4, 7/3, -2/3, 4/15, 2/15, ...
19/3, -3, 14/15, -2/15, ...
-28/3, 59/15, -16/15, ...
199/15, -5, ...
-274/15, ...
etc.
Without the first column, the antidiagonal sums are (-1)^n * A254667(n+1).
The Bernoulli numbers A027641(n)/A027642(n) or A164555(n)/A027642(n) come from A000027. 0 followed by the Bernoulli numbers comes from A001477. a(0)=0 is a choice.

Crossrefs

Cf. A176327/A176289, A172086/A172087, A141459, A176618, A195240, A254667.

Formula

a(2n) = 0. a(2n+1) = A172086(2n), from the main pure Bernoulli twin numbers.
Showing 1-3 of 3 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.