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.

A377530 Expansion of e.g.f. 1/(1 - x * exp(x))^3.

Original entry on oeis.org

1, 3, 18, 141, 1380, 16095, 217458, 3335745, 57225528, 1085066523, 22526087070, 508042140573, 12367076890644, 323130848000727, 9018976230237834, 267789942962863065, 8427492557547704688, 280194087519310655667, 9813332205452943323190, 361109786425470021564021
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2024

Keywords

Crossrefs

Cf. A006153, A377529.
Cf. A377532, A377534.
Cf. A377504.

Programs

  • Mathematica
    nmax=19; CoefficientList[Series[1/(1 - x * Exp[x])^3,{x,0,nmax}],x]Range[0,nmax]! (* Stefano Spezia, Feb 05 2025 *)
  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(k+2, 2)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(k+2,2)/(n-k)!.
a(n) ~ n! * n^2 / (2 * (1+LambertW(1))^3 * LambertW(1)^n). - Vaclav Kotesovec, Oct 31 2024

A377503 E.g.f. satisfies A(x) = 1/(1 - x * exp(x) * A(x))^2.

Original entry on oeis.org

1, 2, 18, 270, 5936, 173330, 6335772, 278724362, 14350790064, 847007698338, 56397332340020, 4182866692785242, 342022887565717800, 30570009715185100082, 2965368922693150575084, 310276298423966343555690, 34834957115496822249510752, 4177193847524372747798263106
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2024

Keywords

Crossrefs

Cf. A295238, A377504, A377528.
Cf. A006013, A364983.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(3*k+1, k)/((k+1)*(n-k)!));

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A364983.
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(3*k+1,k)/( (k+1)*(n-k)! ).

A377528 E.g.f. satisfies A(x) = 1/(1 - x * exp(x) * A(x))^4.

Original entry on oeis.org

1, 4, 60, 1548, 58456, 2930020, 183763704, 13866109012, 1224251041248, 123885272536452, 14140672597851880, 1797709847594145364, 251941291752251706576, 38593132701417704324356, 6415647343472197357272984, 1150373241484390263973203540, 221318733487356013660505462464
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2024

Keywords

Crossrefs

Cf. A295238, A377503, A377504.
Cf. A377526, A377527.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(5*k+3, k)/((k+1)*(n-k)!));

Formula

E.g.f.: B(x)^4, where B(x) is the e.g.f. of A377526.
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(5*k+3,k)/( (k+1)*(n-k)! ).

A377743 E.g.f. satisfies A(x) = exp(x) / (1 - x * A(x))^3.

Original entry on oeis.org

1, 4, 43, 853, 25141, 989581, 48885187, 2910389875, 202958554057, 16233163690537, 1465257396236551, 147359765665925143, 16341437664329027389, 1981169884084699982701, 260701144663332062732491, 37007345616327485166160651, 5637148375602304430334748945, 917186940500490837457393476817
Offset: 0

Views

Author

Seiichi Manyama, Nov 06 2024

Keywords

Crossrefs

Cf. A194471, A377742, A377744.
Cf. A373324, A377504.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (k+1)^(n-k-1)*binomial(4*k+2, k)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..n} (k+1)^(n-k-1) * binomial(4*k+2,k)/(n-k)!.
Showing 1-4 of 4 results.

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

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