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.

A354309 Expansion of e.g.f. 1/(1 - 2*x)^(x/2).

Original entry on oeis.org

1, 0, 2, 6, 44, 360, 3744, 46200, 662864, 10838016, 198943200, 4050937440, 90613710912, 2208677328000, 58265734055424, 1653914478303360, 50263564166365440, 1628300694034022400, 56012708047907510784, 2039053421375533094400, 78314004507947110456320
Offset: 0

Views

Author

Seiichi Manyama, May 23 2022

Keywords

Crossrefs

Cf. A066166, A354310.
Cf. A053491, A354311, A354315, A354319.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[1/(1-2x)^(x/2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 10 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-2*x)^(x/2)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=2, i, j*2^(j-2)/(j-1)*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\2, 2^(n-2*k)*abs(stirling(n-k, k, 1))/(n-k)!);

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=2..n} k * 2^(k-2)/(k-1) * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-2*k) * |Stirling1(n-k,k)|/(n-k)!.
a(n) ~ sqrt(Pi) * 2^(n + 1/2) * n^(n - 1/4) / (Gamma(1/4) * exp(n)). - Vaclav Kotesovec, Mar 14 2024

A354312 Expansion of e.g.f. exp( x/3 * (exp(3 * x) - 1) ).

Original entry on oeis.org

1, 0, 2, 9, 48, 315, 2496, 22491, 223728, 2437371, 28931040, 371291283, 5111412120, 75014135235, 1168157451384, 19228202401635, 333378840718944, 6069073767712587, 115683487658404272, 2303091818149762899, 47784447190060311240, 1031179733234906055507
Offset: 0

Views

Author

Seiichi Manyama, May 23 2022

Keywords

Crossrefs

Cf. A052506, A354311.
Cf. A351734, A351737, A354310, A354314.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x/3*(exp(3*x)-1))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j*3^(j-2)*binomial(i-1, j-1)*v[i-j+1])); v;
  • PARI
    a(n) = n!*sum(k=0, n\2, 3^(n-2*k)*stirling(n-k, k, 2)/(n-k)!);

Formula

a(0) = 1; a(n) = Sum_{k=2..n} k * 3^(k-2) * binomial(n-1,k-1) * a(n-k).
a(n) = n! * Sum_{k=0..floor(n/2)} 3^(n-2*k) * Stirling2(n-k,k)/(n-k)!.

A354316 Expansion of e.g.f. 1/(1 + x/3 * log(1 - 3 * x)).

Original entry on oeis.org

1, 0, 2, 9, 96, 1170, 18324, 340200, 7360128, 181476288, 5024611440, 154319988240, 5206240427904, 191372822989920, 7612497915813504, 325791049256094240, 14925809593280332800, 728828735500650355200, 37786217117138333005824
Offset: 0

Views

Author

Seiichi Manyama, May 23 2022

Keywords

Crossrefs

Cf. A052830, A354315.
Cf. A354310.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[1/(1+x/3 Log[1-3x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 06 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x/3*log(1-3*x))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i!*sum(j=2, i, 3^(j-2)/(j-1)*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\2, 3^(n-2*k)*k!*abs(stirling(n-k, k, 1))/(n-k)!);

Formula

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

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.