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.

Previous Showing 11-15 of 15 results.

A327509 Number of set partitions of [n] where each subset is again partitioned into eight nonempty subsets.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 36, 750, 11880, 159027, 1899612, 20912320, 216627840, 2141770488, 20416870188, 189100389270, 1713143123640, 15314761051669, 137723007972924, 1310008783707360, 14647748873844240, 215375952901752225, 4079250159907459680
Offset: 0

Views

Author

Alois P. Heinz, Sep 14 2019

Keywords

Links

Crossrefs

Column k=8 of A324162.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
      *binomial(n-1, j-1)*Stirling2(j, 8), j=8..n))
    end:
seq(a(n), n=0..27);
  • PARI
    a(n) = sum(k=0, n\8, (8*k)!*stirling(n, 8*k, 2)/(8!^k*k!)); \\ Seiichi Manyama, May 07 2022

Formula

E.g.f.: exp((exp(x)-1)^8/8!).
a(n) = Sum_{k=0..floor(n/8)} (8*k)! * Stirling2(n,8*k)/(8!^k * k!). - Seiichi Manyama, May 07 2022

A327510 Number of set partitions of [n] where each subset is again partitioned into nine nonempty subsets.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 45, 1155, 22275, 359502, 5135130, 67128490, 820784250, 9528822303, 106175420065, 1144618783815, 12011663703975, 123297356170054, 1243260840764910, 12377559175117290, 122870882863640450, 1247553197735599755, 13803307806688911225
Offset: 0

Views

Author

Alois P. Heinz, Sep 14 2019

Keywords

Links

Crossrefs

Column k=9 of A324162.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
      *binomial(n-1, j-1)*Stirling2(j, 9), j=9..n))
    end:
seq(a(n), n=0..27);
  • PARI
    a(n) = sum(k=0, n\9, (9*k)!*stirling(n, 9*k, 2)/(9!^k*k!)); \\ Seiichi Manyama, May 07 2022

Formula

E.g.f.: exp((exp(x)-1)^9/9!).
a(n) = Sum_{k=0..floor(n/9)} (9*k)! * Stirling2(n,9*k)/(9!^k * k!). - Seiichi Manyama, May 07 2022

A327511 Number of set partitions of [n] where each subset is again partitioned into ten nonempty subsets.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 55, 1705, 39325, 752752, 12662650, 193754990, 2758334150, 37112163803, 477297033785, 5917585057033, 71187151690655, 835145968875284, 9593573078823360, 108264887496309962, 1203738001326003000, 13226402531839795155
Offset: 0

Views

Author

Alois P. Heinz, Sep 14 2019

Keywords

Links

Crossrefs

Column k=10 of A324162.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
      *binomial(n-1, j-1)*Stirling2(j, 10), j=10..n))
    end:
seq(a(n), n=0..27);
  • PARI
    a(n) = sum(k=0, n\10, (10*k)!*stirling(n, 10*k, 2)/(10!^k*k!)); \\ Seiichi Manyama, May 07 2022

Formula

E.g.f.: exp((exp(x)-1)^10/10!).
a(n) = Sum_{k=0..floor(n/10)} (10*k)! * Stirling2(n,10*k)/(10!^k * k!). - Seiichi Manyama, May 07 2022

A357868 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).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 0, 2, 13, 0, 1, 0, 0, 6, 75, 0, 1, 0, 0, 6, 38, 541, 0, 1, 0, 0, 0, 36, 270, 4683, 0, 1, 0, 0, 0, 24, 150, 2342, 47293, 0, 1, 0, 0, 0, 0, 240, 1260, 23646, 545835, 0, 1, 0, 0, 0, 0, 120, 1560, 16926, 272918, 7087261, 0, 1, 0, 0, 0, 0, 0, 1800, 8400, 197316, 3543630, 102247563, 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,   3,   2,   0,   0,   0, ...
  0,  13,   6,   6,   0,   0, ...
  0,  75,  38,  36,  24,   0, ...
  0, 541, 270, 150, 240, 120, ...

Crossrefs

Columns k=0-4 give: A000007, A000670, A052841, A353774, A353775.
Cf. A324162, A357293, A357869, A357881.

Programs

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

Formula

For k > 0, e.g.f. of column k: 1/(1 - (exp(x) - 1)^k).
T(0,k) = 1; T(n,k) = k! * Sum_{j=1..n} binomial(n,j) * Stirling2(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).
Previous Showing 11-15 of 15 results.

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