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.

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!.

A353404 Expansion of e.g.f. exp(-log(1 - x)^5).

Original entry on oeis.org

1, 0, 0, 0, 0, 120, 1800, 21000, 235200, 2693880, 34133400, 509823600, 9012207600, 180053908320, 3870208261920, 87083930169600, 2034907999488000, 49491370609706880, 1259740748821328640, 33710658096392887680, 949893399326820528000
Offset: 0

Views

Author

Seiichi Manyama, May 06 2022

Keywords

Links

Crossrefs

Column k=5 of A357882.
Cf. A000142, A353344, A353358.
Cf. A347004, A353200.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[Exp[-Log[1-x]^5],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 13 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-log(1-x)^5)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x)^(-log(1-x)^4)))
  • 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)*abs(stirling(j, 5, 1))*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\5, (5*k)!*abs(stirling(n, 5*k, 1))/k!);

Formula

E.g.f.: (1 - x)^(-(log(1 - x))^4).
a(0) = 1; a(n) = 120 * Sum_{k=1..n} binomial(n-1,k-1) * |Stirling1(k,5)| * a(n-k).
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * |Stirling1(n,5*k)|/k!.

A357881 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)|.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 0, 2, 14, 0, 1, 0, 0, 6, 88, 0, 1, 0, 0, 6, 46, 694, 0, 1, 0, 0, 0, 36, 340, 6578, 0, 1, 0, 0, 0, 24, 210, 3308, 72792, 0, 1, 0, 0, 0, 0, 240, 2070, 36288, 920904, 0, 1, 0, 0, 0, 0, 120, 2040, 24864, 460752, 13109088, 0, 1, 0, 0, 0, 0, 0, 1800, 17640, 310632, 6551424, 207360912, 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,   3,   2,   0,   0,   0, ...
  0,  14,   6,   6,   0,   0, ...
  0,  88,  46,  36,  24,   0, ...
  0, 694, 340, 210, 240, 120, ...

Crossrefs

Columns k=0-5 give: A000007, A007840, A052811, A353118, A353119, A353200.
Cf. A357119, A357868, A357882.

Programs

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

Formula

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

A357883 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)|/(k!^j * j!).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 6, 0, 1, 0, 0, 3, 24, 0, 1, 0, 0, 1, 14, 120, 0, 1, 0, 0, 0, 6, 80, 720, 0, 1, 0, 0, 0, 1, 35, 544, 5040, 0, 1, 0, 0, 0, 0, 10, 235, 4284, 40320, 0, 1, 0, 0, 0, 0, 1, 85, 1834, 38310, 362880, 0, 1, 0, 0, 0, 0, 0, 15, 735, 16352, 383256, 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,  1,  0,  0, 0, ...
  0,   6,  3,  1,  0, 0, ...
  0,  24, 14,  6,  1, 0, ...
  0, 120, 80, 35, 10, 1, ...

Crossrefs

Columns k=0-5 give: A000007, A000142, A347001, A347002, A347003, A347004.
Cf. A324162, A357119, A357881, A357882.

Programs

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

Formula

For k > 0, e.g.f. of column k: exp((-log(1-x))^k / k!).
T(0,k) = 1; T(n,k) = Sum_{j=1..n} binomial(n-1,j-1) * |Stirling1(j,k)| * T(n-j,k).
Showing 1-5 of 5 results.

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