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.

A377826 E.g.f. satisfies A(x) = (1 + x) * exp(x * A(x)).

Original entry on oeis.org

1, 2, 7, 49, 489, 6521, 108643, 2178107, 51084337, 1373054833, 41624314371, 1405311853595, 52299954524953, 2127347522554073, 93902399411048803, 4470613587492385051, 228362858274694209249, 12458393118650371672673, 722983769486947261178371
Offset: 0

Views

Author

Seiichi Manyama, Nov 09 2024

Keywords

Crossrefs

Programs

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

Formula

E.g.f.: (1+x) * exp( -LambertW(-x*(1+x)) ).
E.g.f.: -LambertW(-x*(1+x))/x.
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(k+1,n-k)/k!.
a(n) ~ sqrt(-2*sqrt(1 + 4*exp(-1)) + 2 + 8*exp(-1)) * 2^n * n^(n-1) / ((-1 + sqrt(1 + 4*exp(-1)))^(n+1) * exp(n - 1/2)). - Vaclav Kotesovec, Nov 09 2024

A377829 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x)/(1 + x)^2 ).

Original entry on oeis.org

1, 3, 25, 364, 7713, 216216, 7568041, 318256800, 15644919681, 880848974080, 55912403743161, 3951344780946432, 307737594185310625, 26190457718737019904, 2418475248758250599625, 240846113359411822759936, 25731326615411044591298049, 2935802801104074173428531200
Offset: 0

Views

Author

Seiichi Manyama, Nov 09 2024

Keywords

Crossrefs

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = (1 + x*A(x))^2 * exp(x * A(x)).
a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(2*n+2,n-k)/k!.
a(n) ~ (2*(1 + sqrt(2)))^(n + 1/2) * n^(n-1) / exp((2 - sqrt(2))*n + 1 - sqrt(2)). - Vaclav Kotesovec, Nov 09 2024

A377828 E.g.f. satisfies A(x) = (1 + x)^3 * exp(x * A(x)).

Original entry on oeis.org

1, 4, 21, 193, 2669, 48711, 1113325, 30615019, 984983193, 36319515355, 1510538562641, 69968975169567, 3572684914283941, 199389519518767111, 12075888110164192917, 788850329621989132771, 55289606764547108653361, 4138807268239824817387443, 329564746571982961088975257
Offset: 0

Views

Author

Seiichi Manyama, Nov 09 2024

Keywords

Crossrefs

Programs

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

Formula

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

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 1-4 of 4 results.