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 1-8 of 8 results.

A336951 E.g.f.: 1 / (1 - x * exp(3*x)).

Original entry on oeis.org

1, 1, 8, 69, 780, 11145, 191178, 3823785, 87406056, 2247785073, 64228084110, 2018771719569, 69221032558956, 2571290056399545, 102860527370221026, 4408690840306136505, 201557641172689004112, 9790792086366911655009, 503570143277542340304534
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 08 2020

Keywords

Links

Crossrefs

Column k=3 of A351790.
Cf. A006153, A027471, A235328, A255927, A328182, A336948, A336950, A336952.

Programs

  • Mathematica
    nmax = 18; CoefficientList[Series[1/(1 - x Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[n! Sum[(3 (n - k))^k/k!, {k, 0, n}], {n, 1, 18}]]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k 3^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
  • PARI
    seq(n)={ Vec(serlaplace(1 / (1 - x*exp(3*x + O(x^n))))) } \\ Andrew Howroyd, Aug 08 2020

Formula

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

A328183 Expansion of e.g.f. 1 / (2 - exp(4*x)).

Original entry on oeis.org

1, 4, 48, 832, 19200, 553984, 19181568, 774848512, 35771842560, 1857882947584, 107214340620288, 6805814291464192, 471298297319915520, 35356865248765149184, 2856513752723261227008, 247264693517100223823872, 22830563015939200206766080, 2239752722978295095737974784
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 06 2019

Keywords

Crossrefs

Cf. A000670, A216794, A328182.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n, j)*4^j, j=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 06 2019
  • Mathematica
    nmax = 17; CoefficientList[Series[1/(2 - Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[4^k Binomial[n, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]
Table[2^(2 n - 1) HurwitzLerchPhi[1/2, -n, 0], {n, 0, 17}]

Formula

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

A336948 E.g.f.: 1 / (exp(-3*x) - x).

Original entry on oeis.org

1, 4, 23, 195, 2229, 31863, 546255, 10925757, 249753897, 6422808411, 183524701779, 5768419379913, 197791542799965, 7347180526444359, 293912722687075767, 12597352573293062757, 575928946256877156177, 27976119070974574461363, 1438896686251112024068251
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 08 2020

Keywords

Crossrefs

Cf. A072597, A328182, A336947, A336949, A336951.

Programs

  • Mathematica
    nmax = 18; CoefficientList[Series[1/(Exp[-3 x] - x), {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(3 (n - k + 1))^k/k!, {k, 0, n}], {n, 0, 18}]
a[0] = 1; a[n_] := a[n] = 4 n a[n - 1] - Sum[Binomial[n, k] (-3)^k a[n - k], {k, 2, n}]; Table[a[n], {n, 0, 18}]
  • PARI
    seq(n)={ Vec(serlaplace(1 / (exp(-3*x + O(x*x^n)) - x))) } \\ Andrew Howroyd, Aug 08 2020

Formula

a(n) = n! * Sum_{k=0..n} (3 * (n-k+1))^k / k!.
a(0) = 1; a(n) = 4 * n * a(n-1) - Sum_{k=2..n} binomial(n,k) * (-3)^k * a(n-k).
a(n) ~ n! / ((1 + LambertW(3)) * (LambertW(3)/3)^(n+1)). - Vaclav Kotesovec, Aug 09 2021

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).

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).

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

A354314 Expansion of e.g.f. 1/(1 - x/3 * (exp(3 * x) - 1)).

Original entry on oeis.org

1, 0, 2, 9, 60, 495, 4986, 58401, 780984, 11749779, 196446870, 3612882933, 72484364052, 1575418827879, 36875093680530, 924769734574185, 24737895033896304, 703105981990977915, 21159355356941587470, 672148402091190649629, 22475238194908656800460
Offset: 0

Views

Author

Seiichi Manyama, May 23 2022

Keywords

Crossrefs

Cf. A052848, A354313.
Cf. A288834, A328182, A353999, A354312.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x/3*(exp(3*x)-1))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j*3^(j-2)*binomial(i, j)*v[i-j+1])); v;
  • PARI
    a(n) = n!*sum(k=0, n\2, 3^(n-2*k)*k!*stirling(n-k, k, 2)/(n-k)!);

Formula

a(0) = 1; a(n) = Sum_{k=2..n} k * 3^(k-2) * binomial(n,k) * a(n-k).
a(n) = n! * Sum_{k=0..floor(n/2)} 3^(n-2*k) * k! * Stirling2(n-k,k)/(n-k)!.

A384435 Expansion of e.g.f. 2/(5 - 3*exp(2*x)).

Original entry on oeis.org

1, 3, 24, 282, 4416, 86448, 2030784, 55656912, 1743277056, 61427981568, 2405046994944, 103579443604992, 4866448609591296, 247692476576575488, 13576823521525653504, 797345878311609526272, 49948684871884896731136, 3324530341927517641310208, 234293439367907438337982464
Offset: 0

Views

Author

Seiichi Manyama, Jun 03 2025

Keywords

Links

Crossrefs

Cf. A094417, A326324, A382753.
Cf. A032033, A328182, A384522.
Cf. A201366.

Programs

  • PARI
    a(n) = (-2)^(n+1)*polylog(-n, 5/3)/5;

Formula

a(n) = (-2)^(n+1)/5 * Li_{-n}(5/3), where Li_{n}(x) is the polylogarithm function.
a(n) = 2^(n+1)/5 * Sum_{k>=0} k^n * (3/5)^k.
a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * k! * Stirling2(n,k).
a(n) = (3/5) * A201366(n) = (3/5) * Sum_{k=0..n} 5^k * (-2)^(n-k) * k! * Stirling2(n,k) for n > 0.
a(0) = 1; a(n) = 3 * Sum_{k=1..n} 2^(k-1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 3 * a(n-1) + 5 * Sum_{k=1..n-1} (-2)^(k-1) * binomial(n-1,k) * a(n-k).
Showing 1-8 of 8 results.

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