A081133 -id:A081133

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A081131

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

+10
16

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

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

LINKS

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

FORMULA

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

MATHEMATICA

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

PROG

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

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

CROSSREFS

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).

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Mar 08 2003

STATUS

approved

A053507

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

+10
15

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

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.

REFERENCES

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

LINKS

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

FORMULA

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

MATHEMATICA

Range[0, nn]! CoefficientList[Series[t^3/3!, {x, 0, nn}], x], 1] (* Geoffrey Critzer, Jan 22 2012 *)

Table[Binomial[n-1, 2]n^(n-3), {n, 20}] (* Harvey P. Dale, Sep 24 2019 *)

PROG

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

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

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

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

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

CROSSREFS

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

Equals 2*A065513. A diagonal of A081130.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jan 15 2000

EXTENSIONS

Incorrect Mathematica program deleted by Harvey P. Dale, Sep 24 2019

STATUS

approved

A081132

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

+10
8

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

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

FORMULA

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

EXAMPLE

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.

MAPLE

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}]

PROG

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

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

CROSSREFS

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).

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Mar 08 2003

STATUS

approved

A368849

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

+10
6

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..44.

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

EXAMPLE

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]

MATHEMATICA

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 *)

PROG

(SageMath)

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)])

CROSSREFS

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).

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Jan 11 2024

STATUS

approved

A081196

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Diagonal of A081130.

LINKS

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

FORMULA

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

MAPLE

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

MATHEMATICA

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

PROG

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

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

CROSSREFS

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).

KEYWORD

nonn,easy

AUTHOR

Paul Barry, Mar 11 2003

STATUS

approved

A368982

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

+10
2

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

LINKS

Table of n, a(n) for n=0..44.

FORMULA

T = A369072 - A369025.

EXAMPLE

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]

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 *)

PROG

(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)])

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.

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Jan 11 2024

STATUS

approved

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