A081132 -id:A081132 - 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

A055134 Triangle read by rows: T(n,k) = number of labeled endofunctions on n points with k fixed points.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 8, 12, 6, 1, 81, 108, 54, 12, 1, 1024, 1280, 640, 160, 20, 1, 15625, 18750, 9375, 2500, 375, 30, 1, 279936, 326592, 163296, 45360, 7560, 756, 42, 1, 5764801, 6588344, 3294172, 941192, 168070, 19208, 1372, 56, 1, 134217728

Offset: 0

Christian G. Bower, Apr 25 2000

The same triangle (except for signs) may be obtained from the determinants of the Brahmagupta matrices, setting x->Sqrt[z], y->1, t->n. - Roger L. Bagula, Apr 09 2008
From Bob Selcoe, Nov 15 2014 (Start):
T(n,k)/A000312(n) is the probability P(n,k) that any member (j) of set J={1..n} will be selected k times given n random draws from J. This is equivalent to rolling an n-sided die (with standard assumptions) with sides numbered j=1..n: P(n,k) is the probability that any j will show k times with n rolls.
P(n,k) = (n-2)!*(n-1)^(n-k+1 )/k!*(n-k)!*n^(n-1); n>1. As n approaches infinity, P(n,0) and P(n,1) approach 1/e. (End)
Row sums give n^n (see A000312). - Bob Selcoe, Sep 08 2015

			Triangle T(n,k) begins:
       1;
       0,      1;
       1,      2,      1;
       8,     12,      6,     1;
      81,    108,     54,    12,    1;
    1024,   1280,    640,   160,   20,   1;
   15625,  18750,   9375,  2500,  375,  30,  1;
  279936, 326592, 163296, 45360, 7560, 756, 42, 1;
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Eric W. Weisstein's World of Mathematics, Brahmagupta Matrix.

Columns k=0-2 give: A065440, A055897, A081132(n-2) for n>=2.
Row sums give A000312.
Cf. A055135, A055136.

Clear[B] B[0] = {{x, y}, {t*y, x}}; B[n_] := B[n] = B[n - 1].B[0]; Table[Det[B[n]] /. x -> Sqrt[z] /. y -> 1 /. t -> n, {n, 0, 10}]; a = Join[{{1}}, Table[CoefficientList[Det[B[n]] /. x -> Sqrt[z] /. y ->1 /. t -> n, z], {n, 0, 10}]]; Flatten[a] (* Roger L. Bagula, Apr 09 2008 *)
row[n_] := CoefficientList[(x + n - 1)^n + O[x]^(n+1), x];
Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 13 2017, after Geoffrey Critzer *)
Join[{1, 0, 1}, Table[Binomial[n, k]*(n - 1)^(n - k), {n, 2, 49}, {k, 0, n}]] // Flatten (* G. C. Greubel, Nov 14 2017 *)

for(n=0,15, for(k=0,n, print1(if(n==0 && k==0, 1, if(n==1 && k==0, 0, if(n==1 && k==1, 1, binomial(n,k)*(n-1)^(n-k)))), ", "))) \\ G. C. Greubel, Nov 14 2017

T(n, k) = C(n, k)*(n-1)^(n-k), for n>1.
E.g.f.: (-LambertW(-y)/y)^(x-1)/(1+LambertW(-y)). - Vladeta Jovovic
O.g.f. for row n: (x + n - 1)^n. - Geoffrey Critzer, Mar 21 2010

A081133 a(n) = n^n*binomial(n+2, 2).

Original entry on oeis.org

1, 3, 24, 270, 3840, 65625, 1306368, 29647548, 754974720, 21308126895, 660000000000, 22254310307658, 811365140791296, 31801886192186565, 1333440819066961920, 59553569091796875000, 2822351843277561397248

Offset: 0

Paul Barry, Mar 08 2003

A diagonal of A081130.

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Sequences of the form (n+m)^n*binomial(n+2,2): this sequence (m=0), A081132 (m=1), A081131 (m=2), A053507 (m=3), A081196 (m=4).

[(n^n*Binomial(n+2,2)): n in [0..20]]; // Vincenzo Librandi, Sep 22 2011

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

Join[{1},Table[n^n Binomial[n+2,2],{n,20}]] (* Harvey P. Dale, Dec 27 2011 *)

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

a(n) = n^n*(n+1)*(n+2)/2.

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).

A232055 Triangular array read by rows: T(n,k) is the number of forests of rooted labeled trees such that the vertex labeled with 1 is in a component (rooted tree) of size k, n>=1, 1<=k<=n.

Original entry on oeis.org

1, 1, 2, 3, 4, 9, 16, 18, 27, 64, 125, 128, 162, 256, 625, 1296, 1250, 1440, 1920, 3125, 7776, 16807, 15552, 16875, 20480, 28125, 46656, 117649, 262144, 235298, 244944, 280000, 350000, 489888, 823543, 2097152

Offset: 1

Geoffrey Critzer, Nov 17 2013

Column 1 is A000272.
T(n,n) = A000169(n).
T(n+1,n) = A000312(n).
T(n+2,n)/3 = A081132(n-1).

			1;
1,    2;
3,    4,    9;
16,   18,   27,   64;
125,  128,  162,  256,  625;
1296, 1250, 1440, 1920, 3125, 7776;

Miklos Bona, Introduction to Enumerative Combinatorics, McGraw Hill, 2007, page 282.

Table[Table[Binomial[n,k](k+1)(k+1)^(k-1)(n-k+1)^(n-k-1),{k,0,n}],{n,0,7}]//Grid

T(n,k) = binomial(n-1,k-1)*k^(k-1)*(n-k+1)^(n-k-1).

A353122 Numbers k such that k^k*(k+1) + 1 is prime.

Original entry on oeis.org

0, 1, 2, 3, 6, 9, 186, 198, 8390

Offset: 1

Juri-Stepan Gerasimov, Apr 24 2022

Corresponding primes start 2, 3, 13, 109, 326593, 3874204891, ...
a(9) > 6000. - Jon E. Schoenfield, Jun 05 2022
a(10) > 18000. - Michael S. Branicky, Aug 08 2024

			9 is in the sequence because 9^9*(9+1) + 1 = 3874204891, which is prime.

Cf. A055897, A081132, A108318, A108879, A191513, A191568, A271718.

[n: n in [0..200] | IsPrime(n^n*(n+1) + 1)];

Join[{0}, Select[Range[200], PrimeQ[#^#*(# + 1) + 1] &]] (* Amiram Eldar, Apr 25 2022 *)

isok(k) = ispseudoprime(k^k*(k+1) + 1); \\ Michel Marcus, May 16 2022

a(9) from Michael S. Branicky, Dec 22 2023

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Magma

Magma

Mathematica

PARI

Sage

Formula

A055134 Triangle read by rows: T(n,k) = number of labeled endofunctions on n points with k fixed points.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A081133 a(n) = n^n*binomial(n+2, 2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Sage

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Sage

Formula

A232055 Triangular array read by rows: T(n,k) is the number of forests of rooted labeled trees such that the vertex labeled with 1 is in a component (rooted tree) of size k, n>=1, 1<=k<=n.

Views

Author

Keywords

Comments

Examples

References

Programs

Mathematica

Formula

A353122 Numbers k such that k^k*(k+1) + 1 is prime.

Views

Author