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.

Previous Showing 21-24 of 24 results.

A376515 E.g.f. satisfies A(x) = exp(x^2 * A(x) / (1 - x)).

Original entry on oeis.org

1, 0, 2, 6, 60, 480, 5880, 75600, 1197840, 20865600, 415074240, 9067766400, 218808596160, 5739600746880, 163303845344640, 4998933984844800, 164036362839148800, 5740920215225395200, 213551108122018867200, 8412438143909940940800, 349915152951011468620800
Offset: 0

Views

Author

Seiichi Manyama, Sep 26 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x^2/(1-x)))))
    
  • PARI
    a(n) = n!*sum(k=0, n\2, (k+1)^(k-1)*binomial(n-k-1, n-2*k)/k!);

Formula

E.g.f.: exp( -LambertW(-x^2 / (1-x)) ).
a(n) = n! * Sum_{k=0..floor(n/2)} (k+1)^(k-1) * binomial(n-k-1,n-2*k)/k!.
a(n) ~ exp(1) * sqrt(1 + 4*exp(1) - sqrt(1 + 4*exp(1))) * 2^(n - 1/2) * n^(n-1) / ((1 + 2*exp(1) - sqrt(1 + 4*exp(1))) * (-1 + sqrt(1 + 4*exp(1)))^(n-1)). - Vaclav Kotesovec, Sep 26 2024

A376516 E.g.f. satisfies A(x) = exp(x^3 * A(x) / (1 - x)).

Original entry on oeis.org

1, 0, 0, 6, 24, 120, 1800, 20160, 221760, 3507840, 59875200, 1037836800, 20776694400, 459761702400, 10686605529600, 268901439206400, 7318617546240000, 210804082384896000, 6440850193262284800, 209115023566972723200, 7157303732396353536000, 257535328655939862528000
Offset: 0

Views

Author

Seiichi Manyama, Sep 26 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x^3/(1-x)))))
    
  • PARI
    a(n) = n!*sum(k=0, n\3, (k+1)^(k-1)*binomial(n-2*k-1, n-3*k)/k!);

Formula

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

A376575 E.g.f. A(x) satisfies A(x) = exp(x*A(x)/(1 - x^2)).

Original entry on oeis.org

1, 1, 3, 22, 197, 2376, 35047, 619984, 12772041, 300946816, 7985754251, 235775556864, 7668016756237, 272432946304000, 10499615465565423, 436328344923744256, 19450112299718461073, 925826421005833568256, 46870797202270907609107, 2514801570124507348271104
Offset: 0

Views

Author

Seiichi Manyama, Sep 28 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k+1)^(n-2*k-1) * binomial(n-k-1,k)/(n-2*k)!.
E.g.f.: exp( -LambertW(-x/(1-x^2)) ).
a(n) ~ (1 + 4*exp(-2))^(1/4) * 2^n * n^(n-1) / ((sqrt(1 + 4*exp(-2)) - 1)^n * exp(2*n-1)). - Vaclav Kotesovec, Sep 29 2024

A376576 E.g.f. A(x) satisfies A(x) = exp(x*A(x)/(1 - x^3)).

Original entry on oeis.org

1, 1, 3, 16, 149, 1656, 22567, 372184, 7141689, 156630448, 3871782251, 106504501104, 3227742350197, 106879926110296, 3839600650843791, 148746681984864856, 6181806007303273073, 274355581868776940256, 12951023558423725477459, 647956009961120527442272
Offset: 0

Views

Author

Seiichi Manyama, Sep 28 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n\3, (n-3*k+1)^(n-3*k-1)*binomial(n-2*k-1, k)/(n-3*k)!);
    
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1-x^3)))))

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} (n-3*k+1)^(n-3*k-1) * binomial(n-2*k-1,k)/(n-3*k)!.
E.g.f.: exp( -LambertW(-x/(1-x^3)) ).
From Vaclav Kotesovec, Oct 10 2024: (Start)
E.g.f.: -LambertW(-x/(1-x^3))*(1-x^3)/x.
a(n) ~ sqrt(2^(2/3) * 3^(5/3) / ((2*(9 + sqrt(81 + 12*exp(3))))^(1/3) - 2*exp(1)*(3/(9 + sqrt(81 + 12*exp(3))))^(1/3)) - 2*exp(1)) * 2^(2*n/3) * 3^(4*n/3) * ((9 + sqrt(81 + 12*exp(3)))^(1/3) / (2^(1/3) * (3*(9 + sqrt(81 + 12*exp(3))))^(2/3) - 6*exp(1)))^n * n^(n-1) / exp(n - 1/2). (End)
Previous Showing 21-24 of 24 results.