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.

A088991 Derangement numbers d(n,4) where d(n,k) = k(n-1)(d(n-1,k) + d(n-2,k)), with d(0,k) = 1 and d(1,k) = 0.

Original entry on oeis.org

1, 0, 4, 32, 432, 7424, 157120, 3949056, 114972928, 3805503488, 141137150976, 5797706178560, 261309106499584, 12821127008550912, 680286677982625792, 38814037079505895424, 2369659425449311272960, 154142301601844298776576, 10642813349855965483368448
Offset: 0

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Author

N. J. A. Sloane, Nov 02 2003

Keywords

Links

Crossrefs

Cf. A000166, A053871, A033030, A088992.
Cf. A381504.

Programs

  • Mathematica
    CoefficientList[Series[E^(-x)/(1-4*x)^(1/4), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 08 2013 *)

Formula

Inverse binomial transform of A007696. E.g.f.: exp(-x)/(1-4*x)^(1/4). - Vladeta Jovovic, Dec 17 2003
a(n) ~ n^(n-1/4) * Gamma(3/4) * 4^n / (sqrt(Pi)*exp(n+1/4)). - Vaclav Kotesovec, Oct 08 2013
From Seiichi Manyama, Apr 23 2025: (Start)
E.g.f.: B(x)^4, where B(x) is the e.g.f. of A381504.
a(n) = (-1)^n * n! * Sum_{k=0..n} 4^k * binomial(-1/4,k)/(n-k)!. (End)

A383344 Expansion of e.g.f. exp(-4*x) / (1-x)^4.

Original entry on oeis.org

1, 0, 4, 8, 72, 416, 3520, 31104, 316288, 3525632, 43117056, 572195840, 8191304704, 125761056768, 2060841582592, 35894401335296, 662066514984960, 12890305925218304, 264155723747688448, 5682905054074109952, 128051031032232411136, 3015653024970577018880
Offset: 0

Views

Author

Seiichi Manyama, Apr 23 2025

Keywords

Crossrefs

Column k=4 of A295181.
Cf. A000166, A087981, A137775.
Cf. A088991, A381504.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} (-4)^(n-k) * binomial(k+3,3)/(n-k)!.
a(n) = (n-1) * (a(n-1) + 4*a(n-2)) for n > 1.
E.g.f.: B(x)^4, where B(x) is the e.g.f. of A000166.
a(n) ~ sqrt(2*Pi) * n^(n + 7/2) / (6*exp(n+4)). - Vaclav Kotesovec, Apr 25 2025

A381484 Expansion of e.g.f. exp(-x/3) / (1-3*x)^(1/9).

Original entry on oeis.org

1, 0, 1, 6, 57, 708, 10905, 200538, 4287633, 104507496, 2860291089, 86853807630, 2897638853769, 105357244427244, 4146601837761513, 175632278607964962, 7965651564924845985, 385161391574120046672, 19778647046883844762017, 1074979845580061777989014
Offset: 0

Views

Author

Seiichi Manyama, Apr 23 2025

Keywords

Crossrefs

Cf. A000166, A383313, A381504.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-x/3)/(1-3*x)^(1/9)))

Formula

a(n) = (-1)^n * n! * Sum_{k=0..n} (1/3)^(n-2*k) * binomial(-1/9,k)/(n-k)!.
a(n) = (n-1) * (3*a(n-1) + a(n-2)) for n > 1.
a(n) ~ sqrt(2*Pi) * 3^n * n^(n - 7/18) / (Gamma(1/9) * exp(n + 1/9)). - Vaclav Kotesovec, Apr 23 2025
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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