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.

A375607 Expansion of e.g.f. 1 / (exp(-x^3) - x).

Original entry on oeis.org

1, 1, 2, 12, 72, 480, 3960, 40320, 463680, 5866560, 82857600, 1297296000, 22133865600, 407869862400, 8096683795200, 172405968134400, 3915525770956800, 94443904345190400, 2412049832704512000, 65035187612185190400, 1845812342328514560000
Offset: 0

Views

Author

Seiichi Manyama, Aug 21 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} (n-3*k+1)^k/k!.
a(n) ~ sqrt(2*Pi) * 3^((n+1)/3) * n^(n + 1/2) / ((1 + LambertW(3)) * exp(n) * LambertW(3)^((n+1)/3)). - Vaclav Kotesovec, Aug 21 2024

A375627 Expansion of e.g.f. exp(2*x^2) / (1 - x * exp(x^2)).

Original entry on oeis.org

1, 1, 6, 24, 168, 1260, 11760, 126000, 1545600, 21304080, 326350080, 5497873920, 101048048640, 2011924474560, 43139969832960, 991088998099200, 24286975237324800, 632358338278867200, 17433184834127462400, 507307459608530380800, 15539683835941532467200
Offset: 0

Views

Author

Seiichi Manyama, Aug 21 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k+2)^k/k!.

A375608 Expansion of e.g.f. 1 / (exp(x^2) - x).

Original entry on oeis.org

1, 1, 0, -6, -36, -120, 240, 8400, 82320, 362880, -3507840, -103783680, -1268688960, -4843238400, 175429013760, 5052189542400, 68016191443200, 55329155481600, -23284682272051200, -668640423164313600, -9013925405784499200, 57340797108269875200
Offset: 0

Views

Author

Seiichi Manyama, Aug 21 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (-n+2*k-1)^k/k!.

A375653 Expansion of e.g.f. exp(x^2 + x * exp(x^2)).

Original entry on oeis.org

1, 1, 3, 13, 49, 321, 1891, 13693, 113793, 942049, 9428131, 93837261, 1043918833, 12318170593, 151123106499, 2024107965181, 27672074520961, 406722703307073, 6188268876372163, 98499148173678349, 1645657615850089521, 28317883192163927041
Offset: 0

Views

Author

Seiichi Manyama, Aug 22 2024

Keywords

Crossrefs

Programs

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

Formula

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