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.

A353344 Expansion of e.g.f. exp(-log(1 - x)^3).

Original entry on oeis.org

1, 0, 0, 6, 36, 210, 1710, 17304, 194712, 2402184, 32536080, 481094856, 7703580456, 132658888752, 2443228469136, 47904722262144, 995970495769920, 21879712141853760, 506301721998264000, 12306713585213260800, 313441368701926135680, 8345931596469584686080
Offset: 0

Views

Author

Seiichi Manyama, May 06 2022

Keywords

Links

Crossrefs

Column k=3 of A357882.
Cf. A000142, A353358, A353404.
Cf. A347002, A353118, A353664.

Programs

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

Formula

E.g.f.: (1 - x)^(-(log(1 - x))^2).
a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n-1,k-1) * |Stirling1(k,3)| * a(n-k).
a(n) = Sum_{k=0..floor(n/3)} (3*k)! * |Stirling1(n,3*k)|/k!.

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!.

A357882 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (k*j)!* |Stirling1(n,k*j)|/j!.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 2, 6, 0, 1, 0, 0, 6, 24, 0, 1, 0, 0, 6, 34, 120, 0, 1, 0, 0, 0, 36, 220, 720, 0, 1, 0, 0, 0, 24, 210, 1688, 5040, 0, 1, 0, 0, 0, 0, 240, 1710, 14868, 40320, 0, 1, 0, 0, 0, 0, 120, 2040, 17304, 147684, 362880, 0, 1, 0, 0, 0, 0, 0, 1800, 17640, 194712, 1631376, 3628800, 0
Offset: 0

Views

Author

Seiichi Manyama, Oct 18 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,   6,   6,   6,   0,   0, ...
  0,  24,  34,  36,  24,   0, ...
  0, 120, 220, 210, 240, 120, ...

Crossrefs

Columns k=0-5 give: A000007, A000142, (-1)^n * A009199(n), A353344, A353358, A353404.
Cf. A357119, A357869, A357881.

Programs

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

Formula

For k > 0, e.g.f. of column k: exp((-log(1-x))^k).
T(0,k) = 1; T(n,k) = k! * Sum_{j=1..n} binomial(n-1,j-1) * |Stirling1(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 1-4 of 4 results.

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