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.

A145456 Exponential transform of C(n,6) = A000579.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 7, 28, 84, 210, 462, 1386, 13728, 171171, 1686685, 13461448, 91495768, 551777772, 3142726692, 19787406360, 172188951144, 1999835600301, 24655331721867, 285725747201356, 3034790658153100, 29876851476502030
Offset: 0

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Author

Alois P. Heinz, Oct 10 2008

Keywords

Comments

a(n) is the number of ways of placing n labeled balls into indistinguishable boxes, where in each filled box 6 balls are seen at the top.
a(n) is also the number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains 6 labels.

Links

Crossrefs

6th column of A145460, A143398.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1) *binomial(j,6) *a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);
  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[Exp[x] x^6/6!],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 14 2018 *)

Formula

E.g.f.: exp(exp(x)*x^6/6!).

A145457 Exponential transform of C(n,7) = A000580.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 8, 36, 120, 330, 792, 1716, 5148, 57915, 835120, 9354488, 84047184, 638567124, 4256855760, 25607297880, 144863655024, 869425029957, 7081044528888, 83816629147900, 1131047706331400, 14634713798592030, 173380501913172840
Offset: 0

Views

Author

Alois P. Heinz, Oct 10 2008

Keywords

Comments

a(n) is the number of ways of placing n labeled balls into indistinguishable boxes, where in each filled box 7 balls are seen at the top.
a(n) is also the number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains 7 labels.

Links

Crossrefs

7th column of A145460, A143398.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1) *binomial(j,7) *a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);
  • Mathematica
    a[n_] := a[n] = If[n == 0, 1, Sum[Binomial[n - 1, j - 1] *Binomial[j, 7]* a[n - j], {j, 1, n}]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 11 2018, after Alois P. Heinz *)

Formula

E.g.f.: exp(exp(x)*x^7/7!).

A145458 Exponential transform of C(n,8) = A000581.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 9, 45, 165, 495, 1287, 3003, 6435, 19305, 243100, 3981978, 49959702, 498967170, 4190469570, 30728937690, 201931408074, 1213163827326, 6849350570700, 39615797628550, 296414654550300, 3418235092302030
Offset: 0

Views

Author

Alois P. Heinz, Oct 10 2008

Keywords

Comments

a(n) is the number of ways of placing n labeled balls into indistinguishable boxes, where in each filled box 8 balls are seen at the top.
a(n) is also the number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains 8 labels.

Links

Crossrefs

8th column of A145460, A143398.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1) *binomial(j,8) *a(n-j), j=1..n))
    end:
seq(a(n), n=0..35);
  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[Exp[x] x^8/8!],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Feb 05 2020 *)

Formula

E.g.f.: exp(exp(x)*x^8/8!).

A145459 Exponential transform of C(n,9) = A000582.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 10, 55, 220, 715, 2002, 5005, 11440, 24310, 72930, 1016158, 18643560, 258952330, 2845739820, 26177047270, 209411148144, 1495786618975, 9722602868550, 58373582056075, 329869586346300, 1861266055353705
Offset: 0

Views

Author

Alois P. Heinz, Oct 10 2008

Keywords

Comments

a(n) is the number of ways of placing n labeled balls into indistinguishable boxes, where in each filled box 9 balls are seen at the top.
a(n) is also the number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains 9 labels.

Links

Crossrefs

9th column of A145460, A143398.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1) *binomial(j,9) *a(n-j), j=1..n))
    end:
seq(a(n), n=0..35);

Formula

E.g.f.: exp(exp(x)*x^9/9!).

A355650 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x^k/k! * (exp(x) - 1)).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 0, 2, 5, 1, 0, 0, 3, 15, 1, 0, 0, 3, 16, 52, 1, 0, 0, 0, 6, 65, 203, 1, 0, 0, 0, 4, 10, 336, 877, 1, 0, 0, 0, 0, 10, 105, 1897, 4140, 1, 0, 0, 0, 0, 5, 20, 651, 11824, 21147, 1, 0, 0, 0, 0, 0, 15, 35, 2968, 80145, 115975, 1, 0, 0, 0, 0, 0, 6, 35, 616, 18936, 586000, 678570
Offset: 0

Views

Author

Seiichi Manyama, Jul 12 2022

Keywords

Examples

			Square array begins:
    1,   1,   1,  1,  1, 1, 1, ...
    1,   0,   0,  0,  0, 0, 0, ...
    2,   2,   0,  0,  0, 0, 0, ...
    5,   3,   3,  0,  0, 0, 0, ...
   15,  16,   6,  4,  0, 0, 0, ...
   52,  65,  10, 10,  5, 0, 0, ...
  203, 336, 105, 20, 15, 6, 0, ...

Crossrefs

Columns k=0..3 give A000110, A052506, A354000, A354001.
Cf. A145460, A292892, A355610.

Programs

  • PARI
    T(n, k) = n!*sum(j=0, n\(k+1), stirling(n-k*j, j, 2)/(k!^j*(n-k*j)!));

Formula

T(0,k) = 1 and T(n,k) = ((n-1)!/k!) * Sum_{j=k+1..n} (j/(j-k)!) * T(n-j,k)/(n-j)! for n > 0.
T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} Stirling2(n-k*j,j)/(k!^j * (n-k*j)!).
Previous Showing 11-15 of 15 results.

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