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.

A339058 a(n) = 4^n*Euler(n, 1/4)*2^(valuation_{2}(n + 1)).

Original entry on oeis.org

1, -2, -3, 44, 57, -722, -2763, 196888, 250737, -5746082, -36581523, 2049374444, 7828053417, -259141449842, -2309644635483, 705775346640176, 898621108880097, -38901437271432002, -445777636063460643, 43136210244502819244, 274613643571568682777, -14685255919931552812562
Offset: 0

Views

Author

Peter Luschny, Nov 27 2020

Keywords

Examples

			The array of the general case starts:
[k]
[1] 1,  1,  0, -1,   0,     1,     0,     -17,       0, ... [A198631]
[2] 1,  0, -1,  0,   5,     0,   -61,       0,    1385, ... [A122045]
[3] 1, -1, -2, 13,  22,  -121,  -602,   18581,   30742, ... [A156179]
[4] 1, -2, -3, 44,  57,  -722, -2763,  196888,  250737, ... [this sequence]
[5] 1, -3, -4, 99, 116, -2523, -8764, 1074243, 1242356, ... [A156182]
...

Links

Crossrefs

Cf. A198631, A122045, A156179, A156182, A156191, A339057.
Note the difference from A001586, A188458, and A212435.

Programs

  • Maple
    a := n -> 4^n*euler(n, 1/4)*2^padic[ordp](n+1, 2): seq(a(n), n=0..9);
  • Mathematica
    Array[4^#*EulerE[#, 1/4]*2^IntegerExponent[# + 1, 2] &, 22, 0] (* Michael De Vlieger, Mar 15 2022 *)
  • SageMath
    def euler_sum(n):
    return (-1)^n*sum(2^k*binomial(n, k)*euler_number(k) for k in (0..n))
def a(n): return euler_sum(n) << valuation(n + 1, 2)
print([a(n) for n in range(22)])

A156180 Denominator of Euler(n,1/3).

Original entry on oeis.org

1, 6, 9, 108, 81, 486, 729, 17496, 6561, 39366, 59049, 708588, 531441, 3188646, 4782969, 229582512, 43046721, 258280326, 387420489, 4649045868, 3486784401, 20920706406, 31381059609, 753145430616, 282429536481, 1694577218886, 2541865828329, 30502389939948
Offset: 0

Views

Author

N. J. A. Sloane, Nov 07 2009

Keywords

Examples

			1, -1/6, -2/9, 13/108, 22/81, -121/486, -602/729, 18581/17496, ...

Links

Crossrefs

Cf. A156179.

Programs

  • Maple
    [seq(euler(n,1/3),n=0..50)];
  • Mathematica
    Denominator[EulerE[Range[0,30],1/3]] (* Harvey P. Dale, Aug 14 2011 *)

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)).
Showing 1-3 of 3 results.

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

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