A081131 -id:A081131 - cp's OEIS frontend

A066274 Number of endofunctions of [n] such that 1 is not a fixed point.

0, 2, 18, 192, 2500, 38880, 705894, 14680064, 344373768, 9000000000, 259374246010, 8173092077568, 279577021469772, 10318292052303872, 408700964355468750, 17293822569102704640, 778579070010669895696, 37160496515557841043456, 1874292305362402347591138

Offset: 1

Len Smiley, Dec 09 2001

nonn

a(n) is the number of functional digraphs that are not a solitary rooted tree. - Geoffrey Critzer, Aug 31 2013
For n > 1 a(n) is the number of numbers with n digits in base n. - Gionata Neri, Feb 18 2016
a(n) is the number of pairs of adjacent equal letters in all n-ary words of length n. - John Tyler Rascoe, Nov 19 2024

			a(2)=2: [1->2,2->1], [1->2,2->2].

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Cf. A045531, A066275.

[n^n - n^(n-1): n in [1..20]]; // Vincenzo Librandi, Aug 02 2011

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

a(n) = n^n - n^(n-1).
E.g.f.: T^2/(1-T), where T=T(x) is Euler's tree function (see A000169).
For n > 1 a(n)=1/(Integral_{x=n..infinity} 1/x^n dx). - Francesco Daddi, Aug 01 2011
a(n) = sum(i=1..n-1, C(n,i)*(i^i*(n-i)^(n-i-1))). - Vladimir Kruchinin May 15 2013
E.g.f.: x^2*A''(x) where A(x) is the e.g.f. for A000272. - Geoffrey Critzer, Aug 31 2013
a(n) = 2*A081131(n) = 2*|A070896(n)|. - Geoffrey Critzer, Aug 31 2013

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).

A081130 Square array of binomial transforms of (0,0,1,0,0,0,...), read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 3, 0, 0, 0, 1, 6, 6, 0, 0, 0, 1, 9, 24, 10, 0, 0, 0, 1, 12, 54, 80, 15, 0, 0, 0, 1, 15, 96, 270, 240, 21, 0, 0, 0, 1, 18, 150, 640, 1215, 672, 28, 0, 0, 0, 1, 21, 216, 1250, 3840, 5103, 1792, 36, 0, 0, 0, 1, 24, 294, 2160, 9375, 21504, 20412, 4608, 45, 0

Offset: 0

Paul Barry, Mar 08 2003

Rows, of the square array, are three-fold convolutions of sequences of powers.

			The array begins as:
  0,  0,  0,   0,   0,    0, ...
  0,  0,  0,   0,   0,    0, ...
  0,  1,  1,   1,   1,    1, ... A000012
  0,  3,  6,   9,  12,   15, ... A008585
  0,  6, 24,  54,  96,  150, ... A033581
  0, 10, 80, 270, 640, 1250, ... A244729
The antidiagonal triangle begins as:
  0;
  0, 0;
  0, 0, 0;
  0, 0, 1, 0;
  0, 0, 1, 3,  0;
  0, 0, 1, 6,  6,  0;
  0, 0, 1, 9, 24, 10, 0;

G. C. Greubel, Antidiadoganal rows n = 0..50, flattened

Main diagonal: A081131.
Rows: A000012 (n=2), A008585 (n=3), A033581 (n=4), A244729 (n=5).
Columns: A000217 (k=1), A001788 (k=2), A027472 (k=3), A038845 (k=4), A081135 (k=5), A081136 (k=6), A027474 (k=7), A081138 (k=8), A081139 (k=9), A081140 (k=10), A081141 (k=11), A081142 (k=12), A027476 (k=15).

[k eq n select 0 else (n-k)^(k-2)*Binomial(k,2): k in [0..n], n in [0..12]]; // G. C. Greubel, May 14 2021

Table[If[k==n, 0, (n-k)^(k-2)*Binomial[k, 2]], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 14 2021 *)

T(n, k)=if (k==0, 0, k^(n-2)*binomial(n, 2));
seq(nn) = for (n=0, nn, for (k=0, n, print1(T(k, n-k), ", ")); );
seq(12) \\ Michel Marcus, May 14 2021

flatten([[0 if (k==n) else (n-k)^(k-2)*binomial(k,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 14 2021

T(n, k) = k^(n-2)*binomial(n, 2), with T(n, 0) = 0 (square array).
T(n, n) = A081131(n).
Rows have g.f. x^3/(1-k*x)^n.
From G. C. Greubel, May 14 2021: (Start)
T(k, n-k) = (n-k)^(k-2)*binomial(k,2) with T(n, n) = 0 (antidiagonal triangle).
Sum_{k=0..n} T(n, n-k) = A081197(n). (End)

Term a(5) corrected by G. C. Greubel, May 14 2021

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.

A123744 Circulants of Fibonacci numbers (including F_0 = 0).

Original entry on oeis.org

1, 0, 1, 2, 16, 287, 16128, 2192140, 830952837, 805644641664, 2080690769701456, 14002804169885909807, 247753675148653634781184, 11469641168045182197979378136, 1391545878431673359565624090480585, 442017027765434652128920030338417270784, 367683484076057642925500106042968712221296320

Offset: 0

Wolfdieter Lang, Nov 10 2006, Jan 27 2009

A circulant C_n is the determinant of a circulant n X n matrix M, i.e. one with entries M_{i,j}=a_{i-j} where the indices are taken mod n. Hence C_n=C_n([a_n,a_{n-1},...,a_1]), with the first row of M given.

The eigenvalues of a circulant n X n matrix M(n) are lambda^{(n)}_k=sum(a_j*(rho_n)^(j*k),j=1..n), with the n-th roots of unity (rho_n)^k, k=1..n, where rho_n:=exp(2*Pi/n). See the P. J. Davis reference which uses a different convention.

			n=4: the circular 4 X 4 matrix is M(4) = matrix([[2,1,1,0],[0,2,1,1],[1,0,2,1],[1,1,0,2]]).
n=4: 4th roots of unity: rho_4 = I, (rho_4)^2 = -1, (rho_4)^3 = -I, (rho_4)^4 =1, with I^2=-1. A123744 n=4: the eigenvalues of M(4) are therefore: 0*I^k + 1*(-1)^k + 1*(-I)^k + 2*1^k, k=1,...,4, namely 1-I, 2, 1+I, 4.
n=4: a(4)= Det(M(4)) = 16 = (1-I)*2*(1+I)*4.

P. J. Davis, Circulant Matrices, J. Wiley, New York, 1979.

Cf. A123745 (other Fibonacci circulants without F_0 = 0).
Cf. A081131 (with n instead of Fibonacci(n)).
Cf. A000045.

mm(n) = matdet(matrix(n, n, i, j, fibonacci(n-1-lift(Mod(j-i, n))))); \\ Michel Marcus, Aug 11 2019

a(n) = product(lambda^{(n)}k,k=1..n), with lambda^{(n)}_k=sum(F{j-1}*(rho_n)^(j*k),j=1..n).

a(n) = C_n([F_{n-1},F_{n-2},...,F_0]) with the Fibonacci numbers F_n:=A000045(n) and the circulant C_n (see comment above).

More terms from Michel Marcus, Aug 11 2019

a(0)=1 prepended by Alois P. Heinz, Jun 27 2025

A228273 T(n,k) is the number of s in {1,...,n}^n having longest ending contiguous subsequence with the same value of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 18, 6, 3, 0, 192, 48, 12, 4, 0, 2500, 500, 100, 20, 5, 0, 38880, 6480, 1080, 180, 30, 6, 0, 705894, 100842, 14406, 2058, 294, 42, 7, 0, 14680064, 1835008, 229376, 28672, 3584, 448, 56, 8, 0, 344373768, 38263752, 4251528, 472392, 52488, 5832, 648, 72, 9

Offset: 0

Alois P. Heinz, Aug 19 2013

			T(0,0) = 1: [].
T(1,1) = 1: [1].
T(2,1) = 2: [1,2], [2,1].
T(2,2) = 2: [1,1], [2,2].
T(3,1) = 18: [1,1,2], [1,1,3], [1,2,1], [1,2,3], [1,3,1], [1,3,2], [2,1,2], [2,1,3], [2,2,1], [2,2,3], [2,3,1], [2,3,2], [3,1,2], [3,1,3], [3,2,1], [3,2,3], [3,3,1], [3,3,2].
T(3,2) = 6: [1,2,2], [1,3,3], [2,1,1], [2,3,3], [3,1,1], [3,2,2].
T(3,3) = 3: [1,1,1], [2,2,2], [3,3,3].
Triangle T(n,k) begins:
  1;
  0,        1;
  0,        2,       2;
  0,       18,       6,      3;
  0,      192,      48,     12,     4;
  0,     2500,     500,    100,    20,    5;
  0,    38880,    6480,   1080,   180,   30,   6;
  0,   705894,  100842,  14406,  2058,  294,  42,  7;
  0, 14680064, 1835008, 229376, 28672, 3584, 448, 56,  8;

Alois P. Heinz, Rows n = 0..140, flattened

Row sums give: A000312.
Columns k=0-4 give: A000007, A066274(n) = 2*A081131(n) for n>1, A053506(n) for n>2, A055865(n-1) = A085389(n-1) for n>3, A085390(n-1) for n>4.
Main diagonal gives: A028310.
Lower diagonals include (offsets may differ): A002378, A045991, A085537, A085538, A085539.
Cf. A228154, A228617.

T:= (n, k)-> `if`(n=0 and k=0, 1, `if`(k<1 or k>n, 0,
             `if`(k=n, n, (n-1)*n^(n-k)))):
seq(seq(T(n,k), k=0..n), n=0..12);

f[0,0]=1;
f[n_,k_]:=Which[1<=k<=n-1,n^(n-k)*(n-1),k<1,0,k==n,n,k>n,0];
Table[Table[f[n,k],{k,0,n}],{n,0,10}]//Grid (* Geoffrey Critzer, May 19 2014 *)

T(0,0) = 1, else T(n,k) = 0 for k<1 or k>n, else T(n,n) = n, else T(n,k) = (n-1)*n^(n-k).
Sum_{k=0..n}   T(n,k) = A000312(n).
Sum_{k=0..n} k*T(n,k) = A031972(n).

A303260 Determinant of n X n matrix A[i,j] = (j - i - 1 mod n) + [i=j], i.e., the circulant having (n, 0, 1, ..., n-2) as first row.

Original entry on oeis.org

1, 1, 4, 28, 273, 3421, 52288, 941578, 19505545, 456790123, 11931215316, 343871642632, 10840081272265, 371026432467913, 13702802011918048, 543154131059225686, 23000016472483168305, 1036227971225610466711, 49492629462587441963140, 2497992686980609418282548, 132849300060919364474261281

Offset: 0

M. F. Hasler, Apr 23 2018

nonn

It is remarkable that for odd n, this determinant has its base n+1 digits equal to the middle row: e.g., a(9) = 456790123 is the determinant of the circulant matrix having [4,5,6,7,9,0,1,2,3] as middle row.

a(0) = 1 is (by convention) the determinant of a 0 X 0 matrix.

			a(5) = 3421 is the determinant of the matrix
   ( 5 0 1 2 3 )
   ( 3 5 0 1 2 )
   ( 2 3 5 0 1 )  and 3421 = 23501[6], i.e., written in base 6.
   ( 1 2 3 5 0 )
   ( 0 1 2 3 5 ).

Max Alekseyev, Illustration for a(9) = 456790123 = A219324(20).
N. I. Belukhov, Solution to Problem 14.7 (in Russian), Matematicheskoe Prosveshchenie 15 (2011), pp. 241-244.
Wikipedia, Circulant matrix.

Cf. A081131(n+1) = determinant of the circulant matrix C(n) defined in formula, A070896 (signed variant).

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).

A364709 Triangle read by rows: T(n,k) is the number of forests of labeled rooted hypertrees with n vertices and weight k, 0 <= k < n.

Original entry on oeis.org

1, 2, 1, 9, 9, 1, 64, 96, 28, 1, 625, 1250, 625, 75, 1, 7776, 19440, 14040, 3240, 186, 1, 117649, 352947, 336140, 120050, 14749, 441, 1, 2097152, 7340032, 8716288, 4300800, 870912, 61824, 1016, 1, 43046721, 172186884, 245525742, 156243654, 45605511, 5664330, 245025, 2295, 1

Offset: 1

Paul Laubie, Oct 20 2023

The weight is the number of hypertrees minus 1 plus the weight of each hyperedge which is the number of vertices it connects minus 2.

T(n,k) is also the dimension of the operad ComPreLie in arity n with k commutative products.

			Triangle T(n,k) begins:
n\k   0     1    2    3    4 ...
1     1;
2     2,    1;
3     9,    9,   1;
4    64,   96,  28,   1;
5   625, 1250, 625,  75,   1;
...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Cf. A000169 (k=0), A081131 (k=1).
Row sums are A052888.
Series reversion as e.g.f of A111492 with an offset of 1.

T(n) = my(x='x+O('x^(n+1))); [Vecrev(p) | p<-Vec(serlaplace( serreverse(log(1+x*y)*exp(-x)/y )))]
{my(A=T(10)); for(n=1, #A, print(A[n]))} \\ Andrew Howroyd, Oct 20 2023

E.g.f: series reversion in t of (log(1+x*t)/x)*exp(-t).
T(n,0) = n^(n-1).
T(n,n-1) = 1.

a(23) corrected by Andrew Howroyd, Jan 01 2024

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

A066274 Number of endofunctions of [n] such that 1 is not a fixed point.

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

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A081130 Square array of binomial transforms of (0,0,1,0,0,0,...), read by antidiagonals.

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

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A123744 Circulants of Fibonacci numbers (including F_0 = 0).

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A228273 T(n,k) is the number of s in {1,...,n}^n having longest ending contiguous subsequence with the same value of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A303260 Determinant of n X n matrix A[i,j] = (j - i - 1 mod n) + [i=j], i.e., the circulant having (n, 0, 1, ..., n-2) as first row.

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