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.

A069994 a(n) = Sum_{i=0..2n} B(i)*C(2n+1,i)*6^i where B(i) are the Bernoulli numbers, C(2n,i) the binomial coefficients.

Original entry on oeis.org

-2, 10, -170, 6370, -415826, 41649850, -5922729722, 1134081384850, -281284596509858, 87722769712529770, -33597252908389628234, 15502327024398065811010, -8481855507605264686660850, 5429636257086663655134162970
Offset: 1

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Author

Benoit Cloitre, May 01 2002

Keywords

Comments

Related to those formulas derived from Bernoulli polynomials: Sum_{k>0} sin(k*x)/k^(2n+1) = (-1)^(n+1)/2*x^(2n+1)/(2n+1)!*Sum_{i=0..2n} (2Pi/x)^i*B(i)*C(2n+1,i).

Links

Crossrefs

Cf. A002111, A009843, A069852, A158073.

Programs

  • Maple
    seq(6^(2*n-1)*bernoulli(2*n-1,1/6),n=1..14); # (after Peter Bala) Peter Luschny, Mar 08 2015
  • PARI
    for(n=1,25,print1(sum(i=0,2*n,binomial(2*n+1,i)*bernfrac(i)*6^i),","))

Formula

From Peter Bala, Mar 02 2015: (Start)
a(n) = 6^(2*n - 1)*B(2*n - 1,1/6), where B(n,x) denotes the n-th Bernoulli polynomial. Cf. A002111, A009843, A069852.
Conjecturally, a(n) = 2 * the unsigned numerator of B(2*n - 1,1/6). If true then this sequence is a bisection of 2*A158073.
G.f.: -3*t*sinh(2*t)/sinh(3*t) = -2*t + 10*t^3/3! - 170*t^5/5! + ....
G.f.: Sum_{n >= 0} { 2/(n+1) * Sum_{k = 0..n} (-1)^(k+1)*binomial(n,k)/( (1 - (6*k + 1)*x)*(1 - (6*k + 5)*x) ) } = -2 + 10*x^2 - 170*x^4 + 6370*x^6 - ....
(End)

A158077 Denominator of Bernoulli(n, 1/6).

Original entry on oeis.org

1, 3, 36, 108, 6480, 3888, 326592, 139968, 8398080, 5038848, 665127936, 181398528, 990435962880, 6530347008, 78364164096, 235092492288, 239794342133760, 8463329722368, 13507474236899328, 304679870005248, 201088714203463680
Offset: 0

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Author

N. J. A. Sloane, Nov 08 2009

Keywords

Links

Crossrefs

For numerators see A158073.

Programs

  • Mathematica
    Denominator[BernoulliB[Range[0,20],1/6]] (* Harvey P. Dale, Jan 16 2013 *)

A364112 Expansion of e.g.f. 3*x/(exp(-3*x)+exp(-x)+exp(x)).

Original entry on oeis.org

0, 1, 2, -5, -28, 85, 806, -3185, -41656, 207913, 3428810, -20824925, -413027284, 2961364861, 68560259054, -567040692425, -15005357203312, 140642298254929, 4187120881320338, -43861384856264885, -1450918780756640140, 16798626454194814117, 611263061851828001462, -7751163512199032905505
Offset: 0

Views

Author

F. Chapoton, Jul 13 2023

Keywords

Comments

The terms of even indices are related to Bernoulli numbers. For example, 413027284 = 2^2 * 23 * 73 * 89 * 691 and 15005357203312 = 2^4 * 7 * 31 * 41 * 151 * 193 * 3617.
The terms of odd indices are related to the generalized Bernoulli numbers attached to the primitive Dirichlet character of period 3 (see A002111).

Crossrefs

Very similar to A083007.
Related to A158073 and A002111.

Programs

  • PARI
    my(N=25, x='x+O('x^N)); Vec(serlaplace(3*x/(exp(-3*x)+exp(-x)+exp(x))), -N) \\ Michel Marcus, Jul 13 2023
  • Sage
    x = PowerSeriesRing(QQ, 'x').gen()
N = 20
f = (3*x/((-3*x).exp(N)+(-x).exp(N)+(x).exp(N))).egf_to_ogf()
print(list(f))

Formula

E.g.f.: 3*x/(exp(-3*x)+exp(-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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