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.

A367982 Expansion of e.g.f. exp(-2*x) / (2 - exp(4*x)).

Original entry on oeis.org

1, 2, 36, 584, 13584, 391712, 13563456, 547900544, 25294512384, 1313721631232, 75811987301376, 4812437436975104, 333258221996150784, 25001079178900938752, 2019860245103282896896, 174842541533954981003264, 16143645926877401603702784, 1583744338598987290588086272
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 07 2023

Keywords

Links

Crossrefs

Cf. A000670, A328183, A344037, A355219, A367977, A367979, A367980, A367981, A367983.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!(Laplace( Exp(-2*x)/(2-Exp(4*x)) ))); // G. C. Greubel, Jun 11 2024
  • Mathematica
    nmax = 17; CoefficientList[Series[Exp[-2 x]/(2 - Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (-2)^n + Sum[Binomial[n, k] 4^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]
  • SageMath
    def A367982_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-2*x)/(2-exp(4*x)) ).egf_to_ogf().list()
A367982_list(40) # G. C. Greubel, Jun 11 2024

Formula

a(n) = Sum_{k>=0} (4*k-2)^n / 2^(k+1).
a(n) = (-2)^n + Sum_{k=1..n} binomial(n,k) * 4^k * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * (-2)^(n+k) * A000670(k).

A355218 a(n) = Sum_{k>=1} (3*k - 1)^n / 2^k.

Original entry on oeis.org

1, 5, 43, 557, 9643, 208685, 5419243, 164184557, 5684837803, 221440158125, 9584118542443, 456289689634157, 23698327407870763, 1333388917719691565, 80794290325166308843, 5245268489291712773357, 363231496206350038884523, 26725646191850556128889005, 2082075690178933613292014443
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 24 2022

Keywords

Crossrefs

Cf. A000629, A000670, A007047, A080253, A151919, A328182, A355219, A355220.

Programs

  • Mathematica
    nmax = 18; CoefficientList[Series[Exp[2 x]/(2 - Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2^n + Sum[Binomial[n, k] 3^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

Formula

E.g.f.: exp(2*x) / (2 - exp(3*x)).
a(0) = 1; a(n) = 2^n + Sum_{k=1..n} binomial(n,k) * 3^k * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * 2^(n-k) * 3^k * A000670(k).
a(n) ~ n! * 3^n / (2^(1/3) * log(2)^(n+1)). - Vaclav Kotesovec, Jun 24 2022

A355220 a(n) = Sum_{k>=1} (4*k - 1)^n / 2^k.

Original entry on oeis.org

1, 7, 81, 1399, 32289, 931687, 32259441, 1303134679, 60160827969, 3124574220487, 180312309395601, 11445969681199159, 792626097462398049, 59462922484586318887, 4804064349575887075761, 415847988794676360818839, 38396277196654611908582529, 3766800071614388562865514887
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 24 2022

Keywords

Crossrefs

Cf. A000629, A000670, A080253, A259533, A285067, A328183, A355218, A355219.

Programs

  • Mathematica
    nmax = 17; CoefficientList[Series[Exp[3 x]/(2 - Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 3^n + Sum[Binomial[n, k] 4^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]

Formula

E.g.f.: exp(3*x) / (2 - exp(4*x)).
a(0) = 1; a(n) = 3^n + Sum_{k=1..n} binomial(n,k) * 4^k * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * 3^(n-k) * 4^k * A000670(k).
a(n) ~ n! * 2^(2*n - 1/4) / log(2)^(n+1). - Vaclav Kotesovec, Jun 24 2022
Showing 1-3 of 3 results.

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