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.

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A349280 Irregular triangle read by rows: T(n,k) is the number of arrangements of n labeled children with exactly k rounds; n >= 2, 1 <= k <= floor(n/2).

Original entry on oeis.org

2, 3, 8, 12, 30, 60, 144, 330, 120, 840, 2100, 1260, 5760, 15344, 11760, 1680, 45360, 127008, 113400, 30240, 403200, 1176120, 1169280, 428400, 30240, 3991680, 12054240, 13000680, 5821200, 831600, 43545600, 135508032, 155923680, 80415720, 16632000, 665280
Offset: 2

Views

Author

Steven Finch, Nov 13 2021

Keywords

Comments

A round means the same as a directed ring or circle.

Examples

			Triangle starts:
[2]     2;
[3]     3;
[4]     8,     12;
[5]    30,     60;
[6]   144,    330,    120;
[7]   840,   2100,   1260;
[8]  5760,  15344,  11760,  1680;
[9] 45360, 127008, 113400, 30240;
...
For n = 4, there are 8 ways to make one round and 12 ways to make two rounds.

References

  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999 (Sec. 5.2)

Links

Crossrefs

Row sums give A066166 (Stanley's children's game).
Column 1 gives A001048.
Right border element of row n is A001813(n/2) = |A067994(n)| for even n.
Cf. A008279, A265609, A331327.

Programs

  • Maple
    ser := series((1 - x)^(-x*t), x, 20): xcoeff := n -> coeff(ser, x, n):
T := (n, k) -> n!*coeff(xcoeff(n), t, k):
seq(seq(T(n, k), k = 1..iquo(n,2)), n = 2..12); # Peter Luschny, Nov 13 2021
# second Maple program:
A349280 := (n,k) -> binomial(n,k)*k!*abs(Stirling1(n-k,k)):
seq(print(seq(A349280(n,k), k=1..iquo(n,2))), n=2..12); # Mélika Tebni, May 03 2023
  • Mathematica
    f[k_, n_] := n! SeriesCoefficient[(1 - x)^(-x t), {x, 0, n}, {t, 0, k}]
Table[f[k, n], {n, 2, 12}, {k, 1, Floor[n/2]}]

Formula

G.f.: (1 - x)^(-x*t).
T(n, k) = binomial(n, k)*k!*|Stirling1(n-k, k)|. - Mélika Tebni, May 03 2023
The above formula can also be written as T(n, k) = A008279(n, k)*A331327(n, k) or as T(n, k) = A265609(n + 1, k)*A331327(n, k). - Peter Luschny, May 03 2023

A362834 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = (-1)^n * n! * Sum_{j=0..floor(n/2)} k^j * Stirling1(n-j,j)/(n-j)!.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 3, 0, 1, 0, 6, 6, 20, 0, 1, 0, 8, 9, 64, 90, 0, 1, 0, 10, 12, 132, 300, 594, 0, 1, 0, 12, 15, 224, 630, 2568, 4200, 0, 1, 0, 14, 18, 340, 1080, 6642, 20160, 34544, 0, 1, 0, 16, 21, 480, 1650, 13536, 55440, 193856, 316008, 0
Offset: 0

Views

Author

Seiichi Manyama, May 05 2023

Keywords

Examples

			Square array begins:
  1,  1,   1,   1,    1,    1, ...
  0,  0,   0,   0,    0,    0, ...
  0,  2,   4,   6,    8,   10, ...
  0,  3,   6,   9,   12,   15, ...
  0, 20,  64, 132,  224,  340, ...
  0, 90, 300, 630, 1080, 1650, ...

Crossrefs

Columns k=0..3 give: A000007, A066166, A053489, A053490.
Main diagonal gives A318615.
Cf. A361652.

Programs

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

Formula

E.g.f. of column k: 1/(1 - x)^(k*x).

A349426 Irregular triangle read by rows: T(n,k) is the number of arrangements of n labeled children with exactly k nontrivial rounds; n >= 3, 1 <= k <= floor(n/3).

Original entry on oeis.org

3, 8, 30, 144, 90, 840, 840, 5760, 7280, 45360, 66528, 7560, 403200, 657720, 151200, 3991680, 7064640, 2356200, 43545600, 82285632, 34890240, 1247400, 518918400, 1035365760, 521080560, 43243200, 6706022400, 14013679680, 8034586560, 1059458400
Offset: 3

Views

Author

Steven Finch, Nov 17 2021

Keywords

Comments

A nontrivial round means the same as a ring or circle consisting of more than one child.

Examples

			Triangle starts:
[3]           3;
[4]           8;
[5]          30;
[6]         144,          90;
[7]         840,         840;
[8]        5760,        7280;
[9]       45360,       66528,       7560;
[10]     403200,      657720,     151200;
[11]    3991680,     7064640,    2356200;
[12]   43545600,    82285632,   34890240,    1247400;
[13]  518918400,  1035365760,  521080560,   43243200;
[14] 6706022400, 14013679680, 8034586560, 1059458400;
...
For n = 6, there are 144 ways to make one round and 90 ways to make two rounds.

References

  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999 (Sec. 5.2)

Links

Crossrefs

Row sums give A066165 (variant of Stanley's children's game).
Column 1 gives A001048.
Right border element of row n is A166334(n/3) for each n divisible by 3.
Cf. A066166, A349280 (correspond to Stanley's original game).

Programs

  • Mathematica
    f[k_, n_] := n! SeriesCoefficient[(1 - x)^(-x t) Exp[-x^2 t], {x, 0, n}, {t, 0, k}]
Table[f[k, n], {n, 2, 14}, {k, 1, Floor[n/3]}]

Formula

E.g.f.: (1 - x)^(-x*t) * exp(-x^2*t).

A362837 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = (-1)^n * n! * Sum_{j=0..floor(n/2)} k^(n-j) * Stirling1(n-j,j)/(n-j)!.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 3, 0, 1, 0, 6, 12, 20, 0, 1, 0, 8, 27, 112, 90, 0, 1, 0, 10, 48, 324, 960, 594, 0, 1, 0, 12, 75, 704, 4050, 10848, 4200, 0, 1, 0, 14, 108, 1300, 11520, 64962, 141120, 34544, 0, 1, 0, 16, 147, 2160, 26250, 239616, 1224720, 2122496, 316008, 0
Offset: 0

Views

Author

Seiichi Manyama, May 05 2023

Keywords

Examples

			Square array begins:
  1,  1,   1,    1,     1,     1, ...
  0,  0,   0,    0,     0,     0, ...
  0,  2,   4,    6,     8,    10, ...
  0,  3,  12,   27,    48,    75, ...
  0, 20, 112,  324,   704,  1300, ...
  0, 90, 960, 4050, 11520, 26250, ...

Crossrefs

Columns k=0..3 give: A000007, A066166, A053491, A351735.
Main diagonal gives A362838.
Cf. A362834.

Programs

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

Formula

E.g.f. of column k: 1/(1 - k * x)^x.
Previous Showing 21-24 of 24 results.

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