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 1-4 of 4 results.

A383381 Expansion of e.g.f. exp(-2*x) / (1-x)^5.

Original entry on oeis.org

1, 3, 14, 82, 576, 4688, 43264, 445632, 5062016, 62812288, 844863744, 12239474432, 189939644416, 3142842052608, 55223903596544, 1026805938614272, 20139224002953216, 415503046091767808, 8994794537935765504, 203848794955954716672, 4826475681472562855936, 119162892472107134353408
Offset: 0

Views

Author

Seiichi Manyama, Apr 24 2025

Keywords

Crossrefs

Cf. A001909, A383382, A383383, A383384.
Cf. A000023, A052124, A087981, A383380.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-2*x)/(1-x)^5))

Formula

a(n) = n! * Sum_{k=0..n} (-2)^(n-k) * binomial(k+4,4)/(n-k)!.
a(0) = 1, a(1) = 3; a(n) = (n+2)*a(n-1) + 2*(n-1)*a(n-2).
a(n) ~ sqrt(2*Pi) * n^(n + 9/2) / (24*exp(n+2)). - Vaclav Kotesovec, Apr 25 2025

A383383 Expansion of e.g.f. exp(-4*x) / (1-x)^5.

Original entry on oeis.org

1, 1, 6, 26, 176, 1296, 11296, 110176, 1197696, 14304896, 186166016, 2620022016, 39631568896, 640971452416, 11034441916416, 201411030081536, 3884642996289536, 78929236862140416, 1684881987266215936, 37695662812132212736, 881964287274876665856, 21536903057742987001856
Offset: 0

Views

Author

Seiichi Manyama, Apr 24 2025

Keywords

Crossrefs

Column k=4 of A383341.
Cf. A001909, A383381, A383382, A383384.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-4*x)/(1-x)^5))

Formula

a(n) = n! * Sum_{k=0..n} (-4)^(n-k) * binomial(k+4,4)/(n-k)!.
a(0) = a(1) = 1; a(n) = n*a(n-1) + 4*(n-1)*a(n-2).
a(n) = A383344(n+2)/(4*(n+1)).
a(n) ~ sqrt(2*Pi) * n^(n + 9/2) / (24*exp(n+4)). - Vaclav Kotesovec, Apr 25 2025

A383384 Expansion of e.g.f. exp(-5*x) / (1-x)^5.

Original entry on oeis.org

1, 0, 5, 10, 105, 620, 5725, 52950, 571025, 6686200, 85871925, 1193029250, 17846277625, 285737086500, 4874590170125, 88245858436750, 1689282139310625, 34088182903910000, 723088091207873125, 16083522103093616250, 374280288623526655625, 9093957982779894737500
Offset: 0

Views

Author

Seiichi Manyama, Apr 24 2025

Keywords

Crossrefs

Column k=5 of A295181.
Cf. A001909, A383381, A383382, A383383.
Cf. A000166.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[-5x]/(1-x)^5,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Sep 04 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-5*x)/(1-x)^5))

Formula

a(n) = n! * Sum_{k=0..n} (-5)^(n-k) * binomial(k+4,4)/(n-k)!.
a(n) = (n-1) * (a(n-1) + 5*a(n-2)) for n > 1.
E.g.f.: B(x)^5, where B(x) is the e.g.f. of A000166.
a(n) ~ sqrt(2*Pi) * n^(n + 9/2) / (24*exp(n+5)). - Vaclav Kotesovec, Apr 25 2025

A383378 Expansion of e.g.f. exp(-3*x) / (1-x)^4.

Original entry on oeis.org

1, 1, 5, 21, 129, 897, 7317, 67365, 692577, 7849953, 97199109, 1304688789, 18863836065, 292198665249, 4826470920021, 84669407740773, 1571901715253313, 30786460730863425, 634323280633460613, 13714611211502376597, 310448651226154786881, 7342298348439393120321
Offset: 0

Views

Author

Seiichi Manyama, Apr 24 2025

Keywords

Crossrefs

Column k=3 of A383341.
Cf. A000261, A383344, A383380.
Cf. A010843, A137775, A383382.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-3*x)/(1-x)^4))

Formula

a(n) = n! * Sum_{k=0..n} (-3)^(n-k) * binomial(k+3,3)/(n-k)!.
a(0) = a(1) = 1; a(n) = n*a(n-1) + 3*(n-1)*a(n-2).
a(n) = A137775(n+2)/(3*(n+1)).
a(n) ~ sqrt(2*Pi) * n^(n + 7/2) / (6*exp(n+3)). - Vaclav Kotesovec, Apr 25 2025
Showing 1-4 of 4 results.

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