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 1-5 of 5 results.

A353664 Expansion of e.g.f. exp((exp(x) - 1)^3).

Original entry on oeis.org

1, 0, 0, 6, 36, 150, 900, 9366, 101556, 1031190, 10995300, 134640726, 1844184276, 26656678230, 400614423300, 6347263038486, 106960986110196, 1905688502565270, 35546025523227300, 691014283378745046, 13999772792477879316, 295570215436360196310
Offset: 0

Views

Author

Seiichi Manyama, May 07 2022

Keywords

Crossrefs

Cf. A000110, A052859, A353665.
Cf. A327504, A353344, A353774.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((exp(x)-1)^3)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (3*k)!*x^(3*k)/(k!*prod(j=1, 3*k, 1-j*x))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=6*sum(j=1, i, binomial(i-1, j-1)*stirling(j, 3, 2)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 2)/k!);

Formula

G.f.: Sum_{k>=0} (3*k)! * x^(3*k)/(k! * Product_{j=1..3*k} (1 - j * x)).
a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n-1,k-1) * Stirling2(k,3) * a(n-k).
a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling2(n,3*k)/k!.

A353775 Expansion of e.g.f. 1/(1 - (exp(x) - 1)^4).

Original entry on oeis.org

1, 0, 0, 0, 24, 240, 1560, 8400, 81144, 1638000, 31058520, 482499600, 6905646264, 114015261360, 2456232531480, 59734751403600, 1427946773067384, 33377481440110320, 818549745973204440, 22338800420915168400, 667566534457962216504, 20735588176755396824880
Offset: 0

Views

Author

Seiichi Manyama, May 07 2022

Keywords

Links

Crossrefs

Cf. A000670, A052841, A353774.
Cf. A346895, A353119, A353665.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[1/(1-(Exp[x]-1)^4),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 05 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-(exp(x)-1)^4)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (4*k)!*x^(4*k)/prod(j=1, 4*k, 1-j*x)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=24*sum(j=1, i, binomial(i, j)*stirling(j, 4, 2)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\4, (4*k)!*stirling(n, 4*k, 2));

Formula

G.f.: Sum_{k>=0} (4*k)! * x^(4*k)/Product_{j=1..4*k} (1 - j * x).
a(0) = 1; a(n) = 24 * Sum_{k=1..n} binomial(n,k) * Stirling2(k,4) * a(n-k).
a(n) = Sum_{k=0..floor(n/4)} (4*k)! * Stirling2(n,4*k).
a(n) ~ n! / (8 * log(2)^(n+1)). - Vaclav Kotesovec, May 08 2022

A353358 Expansion of e.g.f. exp(log(1 - x)^4).

Original entry on oeis.org

1, 0, 0, 0, 24, 240, 2040, 17640, 182616, 2340576, 34907520, 567732000, 9811675104, 179804319552, 3507724531584, 72964001073600, 1614757714491456, 37860036000293376, 936291898320463872, 24333527620574701056, 662723505438520771584, 18871765275000834201600
Offset: 0

Views

Author

Seiichi Manyama, May 06 2022

Keywords

Links

Crossrefs

Column k=4 of A357882.
Cf. A000142, A353344, A353404.
Cf. A347003, A353119, A353665.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(log(1-x)^4)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x)^(log(1-x)^3)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=24*sum(j=1, i, binomial(i-1, j-1)*abs(stirling(j, 4, 1))*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\4, (4*k)!*abs(stirling(n, 4*k, 1))/k!);

Formula

E.g.f.: (1 - x)^((log(1 - x))^3).
a(0) = 1; a(n) = 24 * Sum_{k=1..n} binomial(n-1,k-1) * |Stirling1(k,4)| * a(n-k).
a(n) = Sum_{k=0..floor(n/4)} (4*k)! * |Stirling1(n,4*k)|/k!.

A357869 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (k*j)!* Stirling2(n,k*j)/j!.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 2, 5, 0, 1, 0, 0, 6, 15, 0, 1, 0, 0, 6, 26, 52, 0, 1, 0, 0, 0, 36, 150, 203, 0, 1, 0, 0, 0, 24, 150, 962, 877, 0, 1, 0, 0, 0, 0, 240, 900, 6846, 4140, 0, 1, 0, 0, 0, 0, 120, 1560, 9366, 54266, 21147, 0, 1, 0, 0, 0, 0, 0, 1800, 8400, 101556, 471750, 115975, 0
Offset: 0

Views

Author

Seiichi Manyama, Oct 17 2022

Keywords

Examples

			Square array begins:
  1,  1,   1,   1,   1,   1, ...
  0,  1,   0,   0,   0,   0, ...
  0,  2,   2,   0,   0,   0, ...
  0,  5,   6,   6,   0,   0, ...
  0, 15,  26,  36,  24,   0, ...
  0, 52, 150, 150, 240, 120, ...

Links

Crossrefs

Columns k=0-4 give: A000007, A000110, A052859, A353664, A353665.
Cf. A324162, A357293, A357868.

Programs

  • PARI
    T(n, k) = sum(j=0, n, (k*j)!*stirling(n, k*j, 2)/j!);
  • PARI
    T(n, k) = if(k==0, 0^n, n!*polcoef(exp((exp(x+x*O(x^n))-1)^k), n));

Formula

For k > 0, e.g.f. of column k: exp((exp(x) - 1)^k).
T(0,k) = 1; T(n,k) = k! * Sum_{j=1..n} binomial(n-1,j-1) * Stirling2(j,k) * T(n-j,k).

A375773 Expansion of e.g.f. exp((exp(x) - 1)^5).

Original entry on oeis.org

1, 0, 0, 0, 0, 120, 1800, 16800, 126000, 834120, 6917400, 129399600, 3259080000, 72252300120, 1370602233000, 23218349918400, 377834084082000, 6709735404918120, 147369456297228600, 3899127761438053200, 109421543771265852000, 3002806840023201408120
Offset: 0

Views

Author

Seiichi Manyama, Aug 27 2024

Keywords

Crossrefs

Cf. A000110, A052859, A353664, A353665.
Cf. A353404, A373940.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((exp(x)-1)^5)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=120*sum(j=1, i, binomial(i-1, j-1)*stirling(j, 5, 2)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 2)/k!);

Formula

G.f.: Sum_{k>=0} (5*k)! * x^(5*k)/(k! * Product_{j=1..5*k} (1 - j * x)).
a(0) = 1; a(n) = 120 * Sum_{k=1..n} binomial(n-1,k-1) * Stirling2(k,5) * a(n-k).
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * Stirling2(n,5*k)/k!.
Showing 1-5 of 5 results.

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