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 1-7 of 7 results.

A328182 Expansion of e.g.f. 1 / (2 - exp(3*x)).

Original entry on oeis.org

1, 3, 27, 351, 6075, 131463, 3413907, 103429791, 3581223435, 139498558263, 6037616347587, 287444492409231, 14929010774254395, 839982382565841063, 50897213545996785267, 3304312091004451756671, 228821504027595115886955, 16836102104577636004291863, 1311625494765417347634022947
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 06 2019

Keywords

Crossrefs

Cf. A000670, A216794, A247452, A328183.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n, j)*3^j, j=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 06 2019
  • Mathematica
    nmax = 18; CoefficientList[Series[1/(2 - Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[3^k Binomial[n, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
Table[3^n HurwitzLerchPhi[1/2, -n, 0]/2, {n, 0, 18}]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} 3^k * binomial(n,k) * a(n-k).
a(n) = Sum_{k>=0} (3*k)^n / 2^(k + 1).
a(n) = 3^n * A000670(n).
a(n) ~ n! * 3^n / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Aug 09 2021

A367981 Expansion of e.g.f. exp(-x) / (2 - exp(4*x)).

Original entry on oeis.org

1, 3, 41, 699, 16145, 465843, 16129721, 651567339, 30080413985, 1562287110243, 90156154697801, 5722984840599579, 396313048731199025, 29731461242293159443, 2402032174787943537881, 207923994397338180409419, 19198138598327305954291265, 1883400035807194281432757443
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 07 2023

Keywords

Links

Crossrefs

Cf. A000670, A052841, A285067, A328183, A367977, A367979, A367980, A367982, A367983.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!(Laplace( Exp(-x)/(2-Exp(4*x)) ))); // G. C. Greubel, Jun 11 2024
  • Mathematica
    nmax = 17; CoefficientList[Series[Exp[-x]/(2 - Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (-1)^n + Sum[Binomial[n, k] 4^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]
  • SageMath
    def A367981_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-x)/(2-exp(4*x)) ).egf_to_ogf().list()
A367981_list(40) # G. C. Greubel, Jun 11 2024

Formula

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

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).

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).

A336952 E.g.f.: 1 / (1 - x * exp(4*x)).

Original entry on oeis.org

1, 1, 10, 102, 1336, 22200, 443664, 10334128, 275060608, 8236914048, 274069953280, 10031110907136, 400520747437056, 17324601073921024, 807023462798608384, 40278407730378332160, 2144307919689898491904, 121291661335680615284736, 7264376142168665821741056
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 08 2020

Keywords

Links

Crossrefs

Column k=4 of A351790.
Cf. A002697, A006153, A235328, A326324, A328183, A336950, A336951.

Programs

  • Mathematica
    nmax = 18; CoefficientList[Series[1/(1 - x Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[n! Sum[(4 (n - k))^k/k!, {k, 0, n}], {n, 1, 18}]]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k 4^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
  • PARI
    seq(n)={ Vec(serlaplace(1 / (1 - x*exp(4*x + O(x^n))))) } \\ Andrew Howroyd, Aug 08 2020

Formula

a(n) = n! * Sum_{k=0..n} (4 * (n-k))^k / k!.
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k * 4^(k-1) * a(n-k).
a(n) ~ n! * (4/LambertW(4))^n / (1 + LambertW(4)). - Vaclav Kotesovec, Aug 09 2021

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 1-7 of 7 results.

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