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.

A368981 a(n) = Sum_{k=0..n} binomial(n, k - 1)*(1 - k)^(k - 1)*(n - k)*(n - k + 1)^(n - k).

Original entry on oeis.org

0, 0, 2, 12, 168, 1720, 33360, 492324, 12510848, 242010864, 7645282560, 183157788220, 6930019734528, 198083231524776, 8738660263983104, 290276762478721620, 14634486747811184640, 554012204526293864416, 31427811840457845964800, 1335650409538235449288812, 84210181959664202315202560
Offset: 0

Views

Author

Peter Luschny, Jan 11 2024

Keywords

Crossrefs

Cf. A368849.

Programs

  • Mathematica
    A368981[n_] :=Sum[Binomial[n, k-1] If[k == 1, 1, (1-k)^(k-1)] (n-k) (n-k+1)^(n-k), {k, 0, n}];
Array[A368981, 25, 0] (* Paolo Xausa, Jan 13 2024 *)
  • SageMath
    def a(n):
    return sum(binomial(n, k-1)*(1 - k)^(k - 1)*(n - k)*(n - k + 1)^(n - k)
           for k in range(n + 1))
print([a(n) for n in range(0, 21)])

Formula

Alternating row sums of A368849, negated.

A369016 Triangle read by rows: T(n, k) = binomial(n - 1, k - 1) * (k - 1)^(k - 1) * (n - k) * (n - k + 1)^(n - k - 1).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 6, 2, 0, 0, 48, 18, 12, 0, 0, 500, 192, 144, 108, 0, 0, 6480, 2500, 1920, 1620, 1280, 0, 0, 100842, 38880, 30000, 25920, 23040, 18750, 0, 0, 1835008, 705894, 544320, 472500, 430080, 393750, 326592, 0
Offset: 0

Views

Author

Peter Luschny, Jan 12 2024

Keywords

Examples

			Triangle starts:
  [0] [0]
  [1] [0,       0]
  [2] [0,       1,      0]
  [3] [0,       6,      2,      0]
  [4] [0,      48,     18,     12,      0]
  [5] [0,     500,    192,    144,    108,      0]
  [6] [0,    6480,   2500,   1920,   1620,   1280,      0]
  [7] [0,  100842,  38880,  30000,  25920,  23040,  18750,      0]
  [8] [0, 1835008, 705894, 544320, 472500, 430080, 393750, 326592, 0]

Crossrefs

A368849, A368982 and this sequence are alternative sum representation for A001864 with different normalizations.
T(n, k) = A368849(n, k) / n for n >= 1.
T(n, 1) = A053506(n) for n >= 1.
T(n, n - 1) = A055897(n - 1) for n >= 2.
Sum_{k=0..n} T(n, k) = A000435(n) for n >= 1.
Sum_{k=0..n} (-1)^(k+1)*T(n, k) = A368981(n) / n for n >= 1.
Cf. A066320, A369017.

Programs

  • Maple
    T := (n, k) -> binomial(n-1, k-1)*(k-1)^(k-1)*(n-k)*(n-k+1)^(n-k-1):
seq(seq(T(n, k), k = 0..n), n=0..9);
  • Mathematica
    A369016[n_, k_] := Binomial[n-1, k-1] If[k == 1, 1, (k-1)^(k-1)] (n-k) (n-k+1)^(n-k-1); Table[A369016[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 28 2024 *)
  • SageMath
    def T(n, k): return binomial(n-1, k-1)*(k-1)^(k-1)*(n-k)*(n-k+1)^(n-k-1)
for n in range(0, 9): print([T(n, k) for k in range(n + 1)])

Formula

T = B066320 - A369017 (where B066320 = A066320 after adding a 0-column to the left and then setting offset to (0, 0)).

A369019 Triangle read by rows: T(n, k) = binomial(n, k - 1)*(k - 1)^(k - 1)*k*(n - k + 1)^(n - k).

Original entry on oeis.org

0, 0, 1, 0, 2, 4, 0, 9, 12, 36, 0, 64, 72, 144, 432, 0, 625, 640, 1080, 2160, 6400, 0, 7776, 7500, 11520, 19440, 38400, 112500, 0, 117649, 108864, 157500, 241920, 403200, 787500, 2286144, 0, 2097152, 1882384, 2612736, 3780000, 5734400, 9450000, 18289152, 52706752
Offset: 0

Views

Author

Peter Luschny, Jan 13 2024

Keywords

Examples

			Triangle starts:
[0] [0]
[1] [0,      1]
[2] [0,      2,      4]
[3] [0,      9,     12,     36]
[4] [0,     64,     72,    144,    432]
[5] [0,    625,    640,   1080,   2160,   6400]
[6] [0,   7776,   7500,  11520,  19440,  38400, 112500]
[7] [0, 117649, 108864, 157500, 241920, 403200, 787500, 2286144]

Crossrefs

Cf. A369018, A368849.

Programs

  • Maple
    T := (n, k) -> binomial(n, k - 1)*(k - 1)^(k - 1)*k*(n - k + 1)^(n - k):
seq(seq(T(n, k), k = 0..n), n=0..9);
  • Mathematica
    A369019[n_, k_] := Binomial[n, k-1] If[k == 1, 1, (k-1)^(k-1)] k (n-k+1)^(n-k);
Table[A369019[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 27 2024 *)
  • SageMath
    def A369019(n, k):
    return binomial(n, k - 1)*(k - 1)^(k - 1)*k*(n - k + 1)^(n - k)

Formula

T = A369018 - A368849.

A368982 Triangle read by rows: T(n, k) = binomial(n, k - 1) * (k - 1)^(k - 1) * (n - k) * (n - k + 1)^(n - k) / 2.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 9, 3, 0, 0, 96, 36, 24, 0, 0, 1250, 480, 360, 270, 0, 0, 19440, 7500, 5760, 4860, 3840, 0, 0, 352947, 136080, 105000, 90720, 80640, 65625, 0, 0, 7340032, 2823576, 2177280, 1890000, 1720320, 1575000, 1306368, 0
Offset: 0

Views

Author

Peter Luschny, Jan 11 2024

Keywords

Examples

			Triangle starts:
  [0] [0]
  [1] [0,       0]
  [2] [0,       1,       0]
  [3] [0,       9,       3,       0]
  [4] [0,      96,      36,      24,       0]
  [5] [0,    1250,     480,     360,     270,       0]
  [6] [0,   19440,    7500,    5760,    4860,    3840,       0]
  [7] [0,  352947,  136080,  105000,   90720,   80640,   65625,       0]
  [8] [0, 7340032, 2823576, 2177280, 1890000, 1720320, 1575000, 1306368, 0]

Crossrefs

A368849, A369016 and this sequence are alternative sum representation for A001864 with different normalizations.
T(n, k) = A368849(n, k) / 2.
T(n, 1) = A081131(n) for n >= 1.
T(n, n - 1) = A081133(n - 2) for n >= 2.
Sum_{k=0..n} T(n, k) = A036276(n - 1) for n >= 1.
Sum_{k=0..n} (-1)^(k+1)*T(n, k) = A368981(n) / 2 for n >= 0.
Cf. A369072, A369025.

Programs

  • Maple
    T := (n, k) -> binomial(n, k-1)*(k-1)^(k-1)*(n-k)*(n-k+1)^(n-k)/2:
seq(seq(T(n, k), k = 0..n), n=0..9);
  • Mathematica
    A368982[n_, k_] := Binomial[n, k-1] If[k == 1, 1, (k-1)^(k-1)] (n-k) (n-k+1)^(n-k)/2; Table[A368982[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 28 2024 *)
  • SageMath
    def T(n, k): return binomial(n, k-1)*(k-1)^(k-1)*(n-k)*(n-k+1)^(n-k)//2
for n in range(0, 9): print([T(n, k) for k in range(n + 1)])

Formula

T = A369072 - A369025.

A369018 Triangle read by rows: T(n, k) = binomial(n, k - 1)*(k - 1)^(k - 1)*n*(n - k + 1)^(n - k).

Original entry on oeis.org

0, 0, 1, 0, 4, 4, 0, 27, 18, 36, 0, 256, 144, 192, 432, 0, 3125, 1600, 1800, 2700, 6400, 0, 46656, 22500, 23040, 29160, 46080, 112500, 0, 823543, 381024, 367500, 423360, 564480, 918750, 2286144, 0, 16777216, 7529536, 6967296, 7560000, 9175040, 12600000, 20901888, 52706752
Offset: 0

Views

Author

Peter Luschny, Jan 13 2024

Keywords

Examples

			Triangle read by rows:
[0] [0]
[1] [0,      1]
[2] [0,      4,      4]
[3] [0,     27,     18,     36]
[4] [0,    256,    144,    192,    432]
[5] [0,   3125,   1600,   1800,   2700,   6400]
[6] [0,  46656,  22500,  23040,  29160,  46080, 112500]
[7] [0, 823543, 381024, 367500, 423360, 564480, 918750, 2286144]

Crossrefs

Cf. A369019, A368849.

Programs

  • Maple
    T := (n, k) -> binomial(n, k - 1)*(k - 1)^(k - 1)*n*(n - k + 1)^(n - k):
seq(seq(T(n, k), k = 0..n), n=0..9);
  • Mathematica
    A369018[n_, k_] := Binomial[n, k-1] If[k == 1, 1, (k-1)^(k-1)] n (n-k+1)^(n-k);
Table[A369018[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 27 2024 *)
  • SageMath
    def A369018(n, k):
    return binomial(n, k - 1)*(k - 1)^(k - 1)*n*(n - k + 1)^(n - k)

Formula

T = A368849 + A369019.
Showing 1-5 of 5 results.

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