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.

A377964 Expansion of e.g.f. (1+x) * exp(x*(1+x)^3).

Original entry on oeis.org

1, 2, 9, 58, 389, 3186, 29437, 294554, 3233673, 38350594, 484794641, 6522118362, 92857444429, 1390937221298, 21858658599429, 359271578140666, 6156249977141777, 109722278546645634, 2029772196329985433, 38893956306343711994, 770622936760496106261, 15763542538016019828082
Offset: 0

Views

Author

Seiichi Manyama, Nov 12 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} binomial(3*k+1,n-k) / k!.

A377965 Expansion of e.g.f. (1+x)^2 * exp(x*(1+x)^2).

Original entry on oeis.org

1, 3, 11, 55, 309, 1931, 13543, 101991, 828425, 7192819, 66002691, 639830423, 6510397501, 69266297595, 768989536799, 8876171274631, 106301772962193, 1318277355041891, 16892429768517115, 223330116792810999, 3041570471301007301, 42611228176879105003
Offset: 0

Views

Author

Seiichi Manyama, Nov 12 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n, s=2, t=2) = n!*sum(k=0, n, binomial(t*k+s, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} binomial(2*k+2,n-k) / k!.
From Vaclav Kotesovec, Nov 23 2024: (Start)
Recurrence: (n^2 - 3*n + 4)*a(n) = (n^2 - 3*n + 8)*a(n-1) + 2*(n-1)*(2*n^2 - 5*n + 4)*a(n-2) + 3*(n-2)*(n-1)*(n^2 - n + 2)*a(n-3).
a(n) ~ 3^(n/3 - 7/6) * exp(-4/81 + 3^(-7/3)*n^(1/3) + 2*3^(-2/3)*n^(2/3) - 2*n/3) * n^(2*(n+1)/3) * (1 + 5813*3^(1/3)/(4374*n^(1/3))). (End)

A378019 E.g.f. satisfies A(x) = (1+x) * exp( x * (1+x) * A(x) ).

Original entry on oeis.org

1, 2, 9, 79, 957, 15441, 309943, 7468301, 210221385, 6774449185, 246049105131, 9947338595085, 443121311695021, 21568178966624993, 1138938283455953919, 64856665518838006861, 3961941908569940501649, 258453847468153873181889, 17932482767578645884498643
Offset: 0

Views

Author

Seiichi Manyama, Nov 14 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((1+x)*exp(-lambertw(-x*(1+x)^2))))
    
  • PARI
    a(n) = n!*sum(k=0, n, (k+1)^(k-1)*binomial(2*k+1, n-k)/k!);

Formula

E.g.f.: (1+x) * exp( -LambertW(-x * (1+x)^2) ).
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(2*k+1,n-k)/k!.
Showing 1-3 of 3 results.