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

A377827 E.g.f. satisfies A(x) = (1 + x)^2 * exp(x * A(x)).

Original entry on oeis.org

1, 3, 13, 106, 1273, 20226, 402589, 9637902, 269967793, 8666441650, 313793596981, 12653878751526, 562489374836041, 27328756018660266, 1440892788988703821, 81940739770677315646, 4999648556871348611425, 325806859913842861709922, 22584652022005415601772645
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(2*k+2, n-k)/k!);

Formula

E.g.f.: (1+x)^2 * exp( -LambertW(-x*(1+x)^2) ).
E.g.f.: -LambertW(-x*(1+x)^2)/x.
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(2*k+2,n-k)/k!.
a(n) ~ sqrt(1 + 3*r) * n^(n-1) / (exp(n - 1/4) * r^(n + 3/4)), where r = 0.2394629861788505554394435808448... is root of the equation r*(1+r)^2 = exp(-1). - Vaclav Kotesovec, Nov 09 2024

A378016 E.g.f. satisfies A(x) = (1+x) * exp( x * (1+x)^2 * A(x) ).

Original entry on oeis.org

1, 2, 11, 115, 1617, 30241, 701923, 19517975, 633387905, 23513238865, 983268873891, 45750603668815, 2344878934878769, 131285573039583977, 7973124098907905603, 522086636316439329511, 36669284618683152764289, 2750044026126526125774625, 219342360538110975815216323
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)^3))))
    
  • PARI
    a(n) = n!*sum(k=0, n, (k+1)^(k-1)*binomial(3*k+1, n-k)/k!);

Formula

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

A378017 E.g.f. satisfies A(x) = (1+x)^2 * exp( x * (1+x) * A(x) ).

Original entry on oeis.org

1, 3, 15, 148, 2077, 38326, 883369, 24431436, 789531705, 29213730010, 1218401262541, 56566561281616, 2893886178904549, 161768999193007974, 9811122121462081281, 641683497800057913556, 45022670799746182036465, 3373421864644139722767538, 268843153008388446079159573
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)^2*exp(-lambertw(-x*(1+x)^3))))
    
  • PARI
    a(n) = n!*sum(k=0, n, (k+1)^(k-1)*binomial(3*k+2, n-k)/k!);

Formula

E.g.f.: (1+x)^2 * exp( -LambertW(-x * (1+x)^3) ).
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(3*k+2,n-k)/k!.
Showing 1-4 of 4 results.