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.

A367983 Expansion of e.g.f. exp(-3*x) / (2 - exp(4*x)).

Original entry on oeis.org

1, 1, 33, 481, 11457, 329281, 11405793, 460726561, 21270068097, 1104703800961, 63750028379553, 4046761389279841, 280235644230863937, 21023317859012763841, 1698493239420829750113, 147024466409751282556321, 13575133989036437786590977, 1331764937006253524751217921
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 07 2023

Keywords

Links

Crossrefs

Cf. A000670, A328183, A346208, A355220, A367977, A367979, A367980, A367981, A367982.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(-3*x)/(2-Exp(4*x)) ))); // G. C. Greubel, Jun 11 2024
  • Mathematica
    nmax = 17; CoefficientList[Series[Exp[-3 x]/(2 - Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (-3)^n + Sum[Binomial[n, k] 4^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]
  • SageMath
    def A367983_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-3*x)/(2-exp(4*x)) ).egf_to_ogf().list()
A367983_list(40) # G. C. Greubel, Jun 11 2024

Formula

a(n) = Sum_{k>=0} (4*k-3)^n / 2^(k+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).

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

A355219 a(n) = Sum_{k>=1} (4*k - 2)^n / 2^k.

Original entry on oeis.org

1, 6, 68, 1176, 27152, 783456, 27126848, 1095801216, 50589024512, 2627443262976, 151623974601728, 9624874873952256, 666516443992297472, 50002158357801885696, 4039720490206565777408, 349685083067909962039296, 32287291853754803207340032, 3167488677197974581176303616
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 24 2022

Keywords

Crossrefs

Cf. A000629, A000670, A007047, A080253, A285067, A328183, A355218, A355220.

Programs

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

Formula

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

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