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 1-5 of 5 results.

A367977 Expansion of e.g.f. exp(-x) / (2 - exp(2*x)).

Original entry on oeis.org

1, 1, 9, 73, 849, 12241, 211929, 4280473, 98806689, 2565862561, 74035143849, 2349822967273, 81361870604529, 3051889548205681, 123282485663042169, 5335770920836028473, 246332487897909570369, 12083010395805261921601, 627555570373369525058889, 34404109751876393769480073
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 07 2023

Keywords

Links

Crossrefs

Cf. A000670, A052841, A080253, A216794, A344037, A367979, A367980, A367981, A367982, A367983.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 50);
Coefficients(R!(Laplace( Exp(-x)/(2-Exp(2*x)) ))) // G. C. Greubel, Jun 10 2024
  • Mathematica
    nmax = 19; CoefficientList[Series[Exp[-x]/(2 - Exp[2 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (-1)^n + Sum[Binomial[n, k] 2^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
  • SageMath
    def A367977_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-x)/(2-exp(2*x)) ).egf_to_ogf().list()
A367977_list(50) # G. C. Greubel, Jun 10 2024

Formula

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

A367979 Expansion of e.g.f. exp(-x) / (2 - exp(3*x)).

Original entry on oeis.org

1, 2, 22, 278, 4822, 104342, 2709622, 82092278, 2842418902, 110720079062, 4792059271222, 228144844817078, 11849163703935382, 666694458859845782, 40397145162583154422, 2622634244645856386678, 181615748103175019442262, 13362823095925278064444502, 1041037845089466806646007222
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 07 2023

Keywords

Links

Crossrefs

Cf. A000670, A052841, A151919, A328182, A346208, A367977, A367980, A367981, A367982, A367983.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!(Laplace( Exp(-x)/(2-Exp(3*x)) ))); // G. C. Greubel, Jun 11 2024
  • Mathematica
    nmax = 18; CoefficientList[Series[Exp[-x]/(2 - Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (-1)^n + Sum[Binomial[n, k] 3^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
  • SageMath
    def A367979_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-x)/(2-exp(3*x)) ).egf_to_ogf().list()
A367979_list(40) # G. C. Greubel, Jun 11 2024

Formula

a(n) = Sum_{k>=0} (3*k-1)^n / 2^(k+1).
a(n) = (-1)^n + Sum_{k=1..n} binomial(n,k) * 3^k * a(n-k).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 3^k * A000670(k).

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).
Showing 1-5 of 5 results.

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