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.

A367878 Expansion of e.g.f. 1/(1 + 2 * x * log(1-x)).

Original entry on oeis.org

1, 0, 4, 6, 112, 540, 8688, 78960, 1343872, 18162144, 346968000, 6157134720, 134058110976, 2912224423680, 72152130903552, 1839996238429440, 51471401675489280, 1500206702407741440, 46934038380170391552, 1535198134749947965440
Offset: 0

Views

Author

Seiichi Manyama, Dec 03 2023

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0,n\2, 2^k*k!*abs(stirling(n-k, k, 1))/(n-k)!);

Formula

a(0) = 1; a(n) = 2 * n! * Sum_{k=2..n} 1/(k-1) * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/2)} 2^k * k! * |Stirling1(n-k,k)|/(n-k)!.

A375684 Expansion of e.g.f. 1 / (1 - x * log(1 - x)).

Original entry on oeis.org

1, 0, -2, -3, 16, 90, -204, -4200, -5312, 254016, 1586160, -17970480, -294932736, 790115040, 54224747136, 216483714720, -10481294822400, -137535688281600, 1798183916660736, 58769251106526720, -95282580797291520, -23811620975395061760, -203282679617698222080
Offset: 0

Views

Author

Seiichi Manyama, Aug 24 2024

Keywords

Crossrefs

Programs

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

Formula

a(0) = 1; a(n) = -n! * Sum_{k=2..n} 1/(k-1) * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k * k! * |Stirling1(n-k,k)|/(n-k)!.

A375688 Expansion of e.g.f. 1 / (1 + 3 * x * log(1 - x))^(1/3).

Original entry on oeis.org

1, 0, 2, 3, 56, 270, 4824, 44520, 866816, 12195792, 267873120, 5073187680, 126754229568, 2999710359360, 85061489235072, 2400155295632640, 76724104598031360, 2502434971473937920, 89428428468644493312, 3300036525511418327040
Offset: 0

Views

Author

Seiichi Manyama, Aug 24 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+3*x*log(1-x))^(1/3)))
    
  • PARI
    a(n) = n!*sum(k=0, n\2, prod(j=0, k-1, 3*j+1)*abs(stirling(n-k, k, 1))/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (Product_{j=0..k-1} (3*j+1)) * |Stirling1(n-k,k)|/(n-k)!.

A375689 Expansion of e.g.f. 1 / (1 + 3 * x * log(1 - x))^(2/3).

Original entry on oeis.org

1, 0, 4, 6, 136, 660, 13188, 123480, 2584160, 37044000, 855658800, 16536548160, 428924382720, 10358056051200, 302474317729920, 8701780305254400, 284949736641177600, 9464366170599782400, 345224605512559518720, 12956112412535827353600
Offset: 0

Views

Author

Seiichi Manyama, Aug 24 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+3*x*log(1-x))^(2/3)))
    
  • PARI
    a(n) = n!*sum(k=0, n\2, prod(j=0, k-1, 3*j+2)*abs(stirling(n-k, k, 1))/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (Product_{j=0..k-1} (3*j+2)) * |Stirling1(n-k,k)|/(n-k)!.
Showing 1-4 of 4 results.