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 1-2 of 2 results.

A233808 Autosequence preceding A198631(n)/A006519(n+1). Numerators.

Original entry on oeis.org

0, 0, 1, 3, 3, 5, 5, 7, 7, -3, -3, 121, 121, -1261, -1261, 20583, 20583, -888403, -888403, 24729925, 24729925, -862992399, -862992399, 36913939769, 36913939769, -1899853421885, -1899853421885
Offset: 0

Views

Author

Paul Curtz, Dec 16 2013

Keywords

Comments

The fractions are g(n)=0, 0, 1, 3/2, 3/2, 5/4, 5/4, 7/4, 7/4, -3/8, -3/8, 121/8, 121/8, -1261/8, -1261/8, 20583/8, 20583/8, -888403/16, -888403/16,... . The denominators are 1, 1, followed by A053644(n+1).
g(n+2) - g(n+1) = A198631(n)/A006519(n+1).
The corresponding fractions to g(n) are f(n) in A165142(n).
g(n) differences table:
0, 0, 1, 3/2, 3/2, 5/4,
0, 1, 1/2, 0, -1/4, 0,
1, -1/2, -1/2, -1/4, 1/4, 1/2, Euler twin numbers (new),
-3/2, 0, 1/4, 1/2, 1/4, -1,
3/2, 1/4, 1/4, -1/4, -5/4, -5/8,
-5/4, 0, -1/2, -1, 5/8, 13/2, etc.
Like A198631(n)/A006519(n+1),g(n) is an autosequence of the second kind.
If we proceed, here for Euler polynomials, like in A233565 for Bernoulli polynomials, we obtain
1) A133138(n)/A007395(n) (unreduced form) or
2) A233508(n)/A232628(n) (reduced form),the first array in A133135.
The Bernoulli's corresponding fractions to 1) are A193815(n)/(A003056(n) with 1 instead of 0).

Crossrefs

Cf. A051716/A051717, Bernoulli twin numbers.

Programs

  • Mathematica
    max = 27; p[0] = 1; p[n_] := (1 + x)*((1 + x)^(n - 1) + x^(n - 1))/2; t = Table[Coefficient[p[n], x, k], {n, 0, max + 2}, {k, 0, max + 2}]; a[n_] := (-1)^n*Inverse[t][[n, 2]] // Numerator; a[0] = 0; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Jan 11 2016 *)

Formula

a(n) = 0, 0, followed by (-1)^n *A141424(n).

A232628 Denominators of the triangle of polynomial coefficients P(0,x)=1, 2*P(n)=(1+x)*((1+x)^(n-1)+x^(n-1)).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1
Offset: 0

Views

Author

Jean-François Alcover and Paul Curtz, Dec 16 2013

Keywords

Comments

See A233508.

Examples

			1,
1, 1,
2, 2, 1,
2, 2, 1, 1,
2, 1, 1, 2, 1,
2, 2, 1, 1, 1, 1, etc.

Crossrefs

Cf. A233508 (numerators).

Programs

  • Mathematica
    p[n_] := (1+x)*((1+x)^(n-1)+x^(n-1))/2; t[n_, k_] := Coefficient[p[n], x, k] // Denominator; Table[t[n, k], {n, 0, 10 }, {k, 0, n}] // Flatten
Showing 1-2 of 2 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.