A081133 -id:A081133 - cp's OEIS frontend

A081131 a(n) = n^(n-2) * binomial(n,2).

0, 0, 1, 9, 96, 1250, 19440, 352947, 7340032, 172186884, 4500000000, 129687123005, 4086546038784, 139788510734886, 5159146026151936, 204350482177734375, 8646911284551352320, 389289535005334947848, 18580248257778920521728, 937146152681201173795569

Offset: 0

Paul Barry, Mar 08 2003

Main diagonal of A081130.
a(n) is the number of partial functions f: {1,2,...,n} -> {1,2,...,n} that have exactly 2 undefined elements. - Geoffrey Critzer, Feb 08 2012
a(n+1) is the determinant of the circulant matrix having (n-1, n-2, ..., 0) as first row, for n >= 1. See A070896 for a variant, and A303260 for a related sequence. - M. F. Hasler, Apr 23 2018
a(n) is the number of birooted labeled trees on n nodes. - Brendan McKay, May 01 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A070896, A081129, A081130, A303260.

Sequences of the form (n+m)^n*binomial(n+2,2): A081133 (m=0), A081132 (m=1), this sequence (m=2), A053507 (m=3), A081196 (m=4).

[n lt 2 select 0 else n^(n-2)*Binomial(n,2): n in [0..20]]; // G. C. Greubel, May 18 2021

Join[{0},Table[n^(n-2) Binomial[n, 2], {n, 1, 20}]] (* Vladimir Joseph Stephan Orlovsky, Apr 19 2011 *)

[0 if (n<2) else n^(n-2)*binomial(n,2) for n in (0..20)] # G. C. Greubel, May 18 2021

a(0) = a(1) = 0, a(n) = n^(n-2)*binomial(n,2).

E.g.f.: T(x)^2/(2*(1-T(x))) where T(x) is the e.g.f. for A000169. - Geoffrey Critzer, Feb 08 2012

A053507 a(n) = binomial(n-1,2)*n^(n-3).

Original entry on oeis.org

0, 0, 1, 12, 150, 2160, 36015, 688128, 14880348, 360000000, 9646149645, 283787919360, 9098660462034, 315866083233792, 11806916748046875, 472877960873902080, 20205339187128111480, 917543123840934346752, 44131536275846038655193

Offset: 1

N. J. A. Sloane, Jan 15 2000

Number of connected unicyclic simple graphs on n labeled nodes such that the unique cycle has length 3. - Len Smiley, Nov 27 2001

Each simple graph (of this type) corresponds to exactly two 'functional digraphs' counted by A065513.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Prop. 5.3.2.

Vincenzo Librandi, Table of n, a(n) for n = 1..100

Cf. A000169, A053506, A053508, A053509, A081133, A081132.

Equals 2*A065513. A diagonal of A081130.

List([1..20], n-> Binomial(n-1,2)*n^(n-3)); # G. C. Greubel, May 15 2019

[Binomial(n-1,2)*n^(n-3):n in [1..20]]; // Vincenzo Librandi, Sep 22 2011

[Binomial(n-1,2)*n^(n-3): n in [1..20]]; // G. C. Greubel, May 15 2019

nn = 20; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}]; Rest[Range[0, nn]! CoefficientList[Series[t^3/3!, {x, 0, nn}], x]] (* Geoffrey Critzer, Jan 22 2012 *)
Table[Binomial[n-1,2]n^(n-3),{n,20}] (* Harvey P. Dale, Sep 24 2019 *)

vector(20, n, binomial(n-1,2)*n^(n-3)) \\ G. C. Greubel, Jan 18 2017

[binomial(n-1,2)*n^(n-3) for n in (1..20)] # G. C. Greubel, May 15 2019

E.g.f.: -LambertW(-x)^3/3!. - Vladeta Jovovic, Apr 07 2001

A081132 a(n) = (n+1)^n*binomial(n+2,2).

Original entry on oeis.org

1, 6, 54, 640, 9375, 163296, 3294172, 75497472, 1937102445, 55000000000, 1711870023666, 57954652913664, 2120125746145771, 83340051191685120, 3503151123046875000, 156797324626531188736, 7445162356977030877593

Offset: 0

Paul Barry, Mar 08 2003

A diagonal of A081130.

a(n) is the sum of all the fixed points in the set of endofunctions on {1,2,...,n+1}, i.e., the functions f:{1,2,...,n+1} -> {1,2,...,n+1}. - Geoffrey Critzer, Sep 17 2011

			a(1) = 6 because there are four functions from {1,2} into {1,2}: (1*,1) (1*,2*) (2,1) (2,2*) and the fixed points (marked *) sum to 6.

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Sequences of the form (n+m)^n*binomial(n+2,2): A081133 (m=0), this sequence (m=1), A081131 (m=2), A053507 (m=3), A081196 (m=4).

[((n+1)^n*Binomial(n+2,2)): n in [0..20]]; // Vincenzo Librandi, Sep 21 2011

seq((n+1)^n*binomial(n+2,2), n=0..20); # G. C. Greubel, May 18 2021

Mathematica
```
Table[n^n*(n+1)/2,{n,20}]
```

[(n+1)^n*binomial(n+2,2) for n in (0..20)] # G. C. Greubel, May 18 2021

a(n) = (n+1)^n*binomial(n+2,2).

A368849 Triangle read by rows: T(n, k) = binomial(n, k - 1)(k - 1)^(k - 1)(n - k)*(n - k + 1)^(n - k).

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 0, 18, 6, 0, 0, 192, 72, 48, 0, 0, 2500, 960, 720, 540, 0, 0, 38880, 15000, 11520, 9720, 7680, 0, 0, 705894, 272160, 210000, 181440, 161280, 131250, 0, 0, 14680064, 5647152, 4354560, 3780000, 3440640, 3150000, 2612736, 0

Offset: 0

Peter Luschny, Jan 11 2024

A motivation for this triangle was to provide an alternative sum representation for A001864(n) = n! * Sum_{k=0..n-2} n^k/k!. See formula 3 and formula 15 in Riordan and Sloane.

			Triangle starts:
  [0] [0]
  [1] [0,        0]
  [2] [0,        2,       0]
  [3] [0,       18,       6,       0]
  [4] [0,      192,      72,      48,      0]
  [5] [0,     2500,     960,     720,     540,       0]
  [6] [0,    38880,   15000,   11520,    9720,    7680,       0]
  [7] [0,   705894,  272160,  210000,  181440,  161280,  131250,       0]
  [8] [0, 14680064, 5647152, 4354560, 3780000, 3440640, 3150000, 2612736, 0]

John Riordan and N. J. A. Sloane, Enumeration of rooted trees by total height, J. Austral. Math. Soc., vol. 10 pp. 278-282, 1969.

T(n, 1) = A066274(n) for n >= 1.
T(n, 1)/(n - 1) = A000169(n) for n >= 2.
T(n, n - 1) = 2*A081133(n) for n >= 1.
Sum_{k=0..n} T(n, k) = A001864(n).
(Sum_{k=0..n} T(n, k)) / n = A000435(n) for n >= 1.
(Sum_{k=0..n} T(n, k)) * n / 2 = A262973(n) for n >= 1.
(Sum_{k=2..n} T(n, k)) / (2*n) = A057500(n) for n >= 1.
T(n, 1)/(n - 1) + (Sum_{k=2..n} T(n, k)) / (2*n) = A368951(n) for n >= 2.
Sum_{k=0..n} (-1)^(k-1) * T(n, k) = A368981(n).

A368849[n_, k_] := Binomial[n, k-1] If[k == 1, 1, (k-1)^(k-1)] (n-k) (n-k+1)^(n-k);
Table[A368849[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 13 2024 *)

def T(n, k):
    return binomial(n, k - 1)*(k - 1)^(k - 1)*(n - k)*(n - k + 1)^(n - k)
for n in range(0, 9): print([n], [T(n, k) for k in range(n + 1)])

A081196 a(n) = (n+4)^n*binomial(n+2,2).

Original entry on oeis.org

1, 15, 216, 3430, 61440, 1240029, 28000000, 701538156, 19349176320, 583247465515, 19090807228416, 674680957031250, 25614222880669696, 1039980693455123385, 44977604109849722880, 2064633276062972568664

Offset: 0

Paul Barry, Mar 11 2003

Diagonal of A081130.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Sequences of the form (n+m)^n*binomial(n+2,2): A081133 (m=0), A081132 (m=1), A081131 (m=2), A053507 (m=3), this sequence (m=4).

[(n+4)^n*Binomial(n+2,2): n in [0..20]]; // Vincenzo Librandi, Aug 07 2013

seq((n+4)^n*binomial(n+2,2), n=0..20); # G. C. Greubel, May 18 2021

Table[(n+4)^n Binomial[n+2, 2], {n, 0, 30}] (* Vincenzo Librandi, Aug 07 2013 *)

[(n+4)^n*binomial(n+2,2) for n in (0..20)] # G. C. Greubel, May 18 2021

a(n) = (n+4)^n*binomial(n+2,2).

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

Peter Luschny, Jan 11 2024

			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]

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.

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);

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 *)

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)])

T = A369072 - A369025.

cp's OEIS Frontend

A081131 a(n) = n^(n-2) * binomial(n,2).

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A053507 a(n) = binomial(n-1,2)*n^(n-3).

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A081132 a(n) = (n+1)^n*binomial(n+2,2).

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A368849 Triangle read by rows: T(n, k) = binomial(n, k - 1)*(k - 1)^(k - 1)*(n - k)*(n - k + 1)^(n - k).

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A081196 a(n) = (n+4)^n*binomial(n+2,2).

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Author

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Crossrefs

Programs

Magma

Maple

Mathematica

Sage

Formula

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

SageMath

Formula

A368849 Triangle read by rows: T(n, k) = binomial(n, k - 1)(k - 1)^(k - 1)(n - k)*(n - k + 1)^(n - k).