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.

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A279172 The autosequence of the first kind between A226158(n) and A278331(n).

Original entry on oeis.org

0, 1, 1, -1, -3, 3, 17, -17, -155, 155, 2073, -2073, -38227, 38227, 929569, -929569, -28820619, 28820619, 1109652905, -1109652905, -51943281731, 51943281731, 2905151042481, -2905151042481, -191329672483963, 191329672483963, 14655626154768697, -14655626154768697
Offset: 0

Views

Author

Jean-François Alcover and Paul Curtz, Dec 07 2016

Keywords

Comments

Peter Luschny introduced the extended Genocchi numbers A226158(n), an autosequence of the first kind. They are linked to the second Bernoulli numbers B+(n) = A164555(n)/A027642(n). Here + is an exponent.
This yields the possible Genocchi twin numbers: -1, -1 followed by a(n).

Links

Crossrefs

Cf. A000918, A027642, A036968, A051716, A164555, A226158, A278331.

Programs

  • Mathematica
    a[n_] := (n+2)*EulerE[n+1, 0] - (n+1)*EulerE[n, 0]; a[0] = 0; Table[a[n], {n, 0, 30}] (* or: *)
Join[{0}, Array[#*EulerE[#-1, 0]&, 32] // Differences // Rest]

Formula

a(n) = A226158(n+2) - A226158(n+1).
a(n) = 2*(2^(n+1) -1)*B+(n+1) -2*(2^(n+2) -1)*B+(n+2).

A364199 Expansion of e.g.f. 2*x/(exp(-2*x)+exp(x)).

Original entry on oeis.org

0, 1, 1, -6, -13, 110, 363, -4214, -18581, 276678, 1525355, -27753022, -183611829, 3948004606, 30473073547, -756031185030, -6669149100757, 187521633674294, 1860949703300139, -58481734930175438, -644853406058229365, 22398157925324204142, 271672536688626976331, -10334883450918076967446
Offset: 0

Views

Author

F. Chapoton, Jul 13 2023

Keywords

Comments

The terms with even indices are related to Bernoulli numbers. For example, 183611829 = 3 * 23 * 691 * 3851 and 6669149100757 = 11^2 * 13 * 257 * 3617 * 4561.
The terms with odd indices are related to the generalized Bernoulli numbers attached to the primitive Dirichlet character of period 3 (see A002111).

Crossrefs

Very similar to the Genocchi numbers A036968.
Related to A156179 and A002111.

Programs

  • PARI
    my(N=25, x='x+O('x^N)); Vec(serlaplace(2*x/(exp(-2*x)+exp(x))), -N) \\ Michel Marcus, Jul 15 2023
  • Sage
    x = PowerSeriesRing(QQ, 'x').gen()
N = 20
f = (2*x/((-2*x).exp(N)+(x).exp(N))).egf_to_ogf()
print(list(f))

Formula

E.g.f.: 2*x/(exp(-2*x)+exp(x)).
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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