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 1-3 of 3 results.

A375945 Expansion of e.g.f. 1 / (1 + 2 * log(1 - x))^(3/2).

Original entry on oeis.org

1, 3, 18, 156, 1758, 24342, 399480, 7577700, 163090500, 3926104860, 104520733560, 3048811591680, 96695722690200, 3312942954681240, 121938065727180480, 4798400761979259120, 201030443703421854480, 8933622147642363338160, 419725992843354254228640
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2024

Keywords

Crossrefs

Programs

  • Mathematica
    nmax=18; CoefficientList[Series[1 / (1 + 2 * Log[1 - x])^(3/2),{x,0,nmax}],x]*Range[0,nmax]! (* Stefano Spezia, Sep 03 2024 *)
  • PARI
    a001147(n) = prod(k=0, n-1, 2*k+1);
    a(n) = sum(k=0, n, a001147(k+1)*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} A001147(k+1) * |Stirling1(n,k)|.
a(n) ~ n^(n+1) / (exp(n/2) * (exp(1/2) - 1)^(n + 3/2)). - Vaclav Kotesovec, Sep 06 2024

A375987 Expansion of e.g.f. (1 + 2 * log(1 - x))^(3/2).

Original entry on oeis.org

1, -3, 0, 6, 42, 318, 2892, 31944, 424596, 6682740, 122318928, 2559121128, 60275236392, 1577894836248, 45427570253712, 1425885338250432, 48443767097018256, 1770703320887526096, 69273368628184075392, 2887794188011931364576, 127778992241790634125984
Offset: 0

Views

Author

Seiichi Manyama, Sep 05 2024

Keywords

Crossrefs

Programs

  • Maple
    A375987 := proc(n)
        add(mul(2*j-3,j=0..k-1)*abs(stirling1(n,k)),k=0..n) ;
    end proc:
    seq(A375987(n),n=0..30) ; # R. J. Mathar, Sep 06 2024
  • PARI
    a(n) = sum(k=0, n, prod(j=0, k-1, 2*j-3)*abs(stirling(n, k, 1)));

Formula

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

A375990 Expansion of e.g.f. (1 + 2 * log(1 - x))^2.

Original entry on oeis.org

1, -4, 4, 16, 64, 304, 1712, 11232, 84384, 715392, 6761088, 70513920, 804683520, 9975536640, 133513989120, 1919012014080, 29482606540800, 482183099596800, 8364495012249600, 153406409645260800, 2965940772905779200, 60291976261386240000
Offset: 0

Views

Author

Seiichi Manyama, Sep 05 2024

Keywords

Crossrefs

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[(1+2Log[1-x])^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 05 2025 *)
  • PARI
    a(n) = if(n==0, 1, -4*(n-1)!+8*abs(stirling(n, 2, 1)));

Formula

a(n) = -4 * (n-1)! + 8 * |Stirling1(n,2)| for n > 0.
Showing 1-3 of 3 results.