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.

A367980 Expansion of e.g.f. exp(-2*x) / (2 - exp(3*x)).

Original entry on oeis.org

1, 1, 19, 217, 3835, 82801, 2150659, 65156617, 2256029515, 87878584801, 3803459964499, 181078683329017, 9404687464288795, 529155742667806801, 32063235363798322339, 2081586179439325213417, 144148514796485770141675, 10606079719868369436964801, 826272285216863547170504179
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 07 2023

Keywords

Links

Crossrefs

Cf. A000670, A328182, A344037, A355218, A367977, A367979, A367981, A367982, A367983.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!(Laplace( Exp(-2*x)/(2-Exp(3*x)) ))); // G. C. Greubel, Jun 11 2024
  • Mathematica
    nmax = 18; CoefficientList[Series[Exp[-2 x]/(2 - Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (-2)^n + Sum[Binomial[n, k] 3^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
  • SageMath
    def A367980_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-2*x)/(2-exp(3*x)) ).egf_to_ogf().list()
A367980_list(40) # G. C. Greubel, Jun 11 2024

Formula

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

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

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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