A081131 -id:A081131 - cp's OEIS frontend

A185025 Triangular array read by rows: T(n,k) is the number of functions f:{1,2,...,n} -> {1,2,...,n} that have exactly k 2-cycles for n >= 0 and 0 <= k <= floor(n/2).

1, 1, 3, 1, 18, 9, 163, 90, 3, 1950, 1100, 75, 28821, 16245, 1575, 15, 505876, 283122, 33810, 735, 10270569, 5699932, 780150, 26460, 105, 236644092, 130267440, 19615932, 884520, 8505, 6098971555, 3332614725, 538325550, 29619450, 467775, 945

Offset: 0

Geoffrey Critzer, Dec 24 2012

It appears that as n gets large, row n conforms to a Poisson distribution with mean = 1/2. In other words, as n gets large, T(n,k) approaches n^n/(2^k*k!*e^(1/2)).

			Triangle begins:
           1;
           1;
           3,          1;
          18,          9;
         163,         90,         3;
        1950,       1100,        75;
       28821,      16245,      1575,       15;
      505876,     283122,     33810,      735;
    10270569,    5699932,    780150,    26460,    105;
   236644092,  130267440,  19615932,   884520,   8505;
  6098971555, 3332614725, 538325550, 29619450, 467775, 945;
  ...

Column k=0 gives A089466.

Cf. A000169, A081131.

nn=10;t=Sum[n^(n-1)x^n/n!,{n,1,nn}]; Range[0,nn]! CoefficientList[Series[Exp[t^2/2(y-1)]/(1-t), {x,0,nn}], {x,y}]//Grid

E.g.f.: exp((T(x)^2/2)*(y-1))/(1 - T(x)) where T(x) is the e.g.f. for A000169.

Sum_{k=1..floor(n/2)} k * T(n,k) = A081131(n).

A225213 Triangular array read by rows. T(n,k) is the number of cycles in the digraph representation of all functions f:{1,2,...,n}->{1,2,...,n} that have length k; 1<=k<=n.

Original entry on oeis.org

1, 4, 1, 27, 9, 2, 256, 96, 32, 6, 3125, 1250, 500, 150, 24, 46656, 19440, 8640, 3240, 864, 120, 823543, 352947, 168070, 72030, 24696, 5880, 720, 16777216, 7340032, 3670016, 1720320, 688128, 215040, 46080, 5040

Offset: 1

Geoffrey Critzer, May 01 2013

Row sums = A190314(n)
  Sum_{k=1..n} T(n,k)*k = A063169(n)
  T(n,n) = (n-1)!
  Column 1 = n^n = A000312
  Column 2 = A081131

			1,
4,      1,
27,     9,      2,
256,    96,     32,     6,
3125,   1250,   500,    150,   24,
46656,  19440,  8640,   3240,  864,   120,
823543, 352947, 168070, 72030, 24696, 5880, 720

Table[Table[(j-1)!Binomial[n,j]n^(n-j),{j,1,n}],{n,1,8}]//Grid

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

E.g.f. for column k: A(x)^k/k * B(x) where A(x) is e.g.f. for A000169 and B(x) is e.g.f. for A000312.

A345632 Sum of terms of even index in the binomial decomposition of n^(n-1).

Original entry on oeis.org

1, 1, 5, 28, 353, 3376, 66637, 908608, 24405761, 432891136, 14712104501, 321504185344, 13218256749601, 343360783937536, 16565151205544957, 498676704524517376, 27614800115689879553, 945381827279671853056, 59095217374989483261925, 2267322327322331161821184, 157904201452248753415276001

Offset: 1

Olivier Gérard, Jun 21 2021

When writing n^(n-1) (A000169) as a sum of powers of n using the binomial theorem, one can separately sum the even and the odd powers of n. This is the even part.

Cf. A345633 (odd part).
Cf. A062024, A302583.
Cf. A000169, A007778, A092364, A081131.

Table[Plus @@ Table[(n-1)^(2 k) Binomial[n-1, 2 k], {k, 0, Floor[n/2]}], {n, 1, 21}]

a(n+1) = Sum_{k=0..floor(n/2)} n^(2k) binomial(n, 2k).

a(n+1) = ((1 - n)^n + (1 + n)^n)/2. - Stefano Spezia, Jun 21 2021

A345633 Sum of terms of odd index in the binomial decomposition of n^(n-1).

Original entry on oeis.org

0, 1, 4, 36, 272, 4400, 51012, 1188544, 18640960, 567108864, 11225320100, 421504185344, 10079828372880, 450353989316608, 12627774819845668, 654244800082329600, 21046391759976988928, 1240529732459024678912, 45032132922921758270916, 2975557672677668838178816

Offset: 1

Olivier Gérard, Jun 21 2021

When writing n^(n-1) (A000169) as a sum of powers of n using the binomial theorem, one can separately sum the even and the odd powers of n. This is the odd part. See the Formula section.

Cf. A345632 (even part).
Cf. A062024, A302583.
Cf. A000169, A007778, A092364, A081131.

Table[Plus @@ Table[(n - 1)^(2 k + 1) Binomial[n - 1, 2 k + 1], {k, 0, Floor[(n - 1)/2]}], {n, 1, 21}]

a(n+1) = Sum_{k=0..floor((n-1)/2)} n^(2k+1)*binomial(n, 2k+1).

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

A383036 The determinant of the matrix representing a totally anti-symmetric quasigroup of order 2*n+1.

Original entry on oeis.org

0, 9, 1250, 352947, 172186884, 129687123005, 139788510734886, 204350482177734375, 389289535005334947848, 937146152681201173795569, 2782184294469515486371964010, 9986310782535957929474146174619, 42632564145606011152267456054687500, 213501642487388555901009081409220318757

Offset: 0

Darío Clavijo, May 21 2025

A totally antisymmetric quasigroup of order 2*n+1 is constructed in a way such that M[i][j] != M[j][i] for i!=j with m = 2*n+1, k = 2 and M[j][i] = k*(j-i) mod m for 0 <= j,i < m.
For any k != 0 mod m the resulting matrix M has the same determinant for each n.
Also the resulting matrix M is circulant and a Latin square.

			For n = 1, a(1) = 9 because:
The resulting totally anti-symetric quasigroup has a matrix:
with k = 1:
  0, 1, 2,
  2, 0, 1,
  1, 2, 0
which has a determinant: 9.
with k = 2:
  0, 2, 1,
  1, 0, 2,
  2, 1, 0
has also the same determinant 9.

Paolo Xausa, Table of n, a(n) for n = 0..100
H. Michael Damm, Totally anti-symmetric quasigroups for all orders n not equal to 2 or 6, Discrete Math., 307:6 (2007), 715-729.
Wikipedia, Quasigroups

Cf. A005408, A081131, A375584.

A383036[n_] := n*(2*n+1)^(2*n); Array[A383036, 15, 0] (* Paolo Xausa, May 28 2025 *)

a(n) = n*(2*n+1)^(2*n) = A081131(2*n+1).

A249632 Triangular array read by rows. T(n,k) is the number of labeled trees with black and white nodes having exactly k black nodes, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 9, 9, 3, 16, 64, 96, 64, 16, 125, 625, 1250, 1250, 625, 125, 1296, 7776, 19440, 25920, 19440, 7776, 1296, 16807, 117649, 352947, 588245, 588245, 352947, 117649, 16807, 262144, 2097152, 7340032, 14680064, 18350080, 14680064, 7340032, 2097152, 262144

Offset: 0

Geoffrey Critzer, Nov 02 2014

Row sums = A038058.
T(n,n) = T(n,0) = n^(n-2) free trees A000272.
T(n,n-1) = T(n,1) = n^(n-1) rooted trees A000169.
T(n,2) = A081131.

			1,
1,    1,
1,    2,    1,
3,    9,    9,     3,
16,   64,   96,    64,    16,
125,  625,  1250,  1250,  625,   125,
1296, 7776, 19440, 25920, 19440, 7776, 1296

F. Harary and E. Palmer, Graphical Enumeration, Academic Press,1973, page 30, exercise 1.10.

nn = 6; f[x_] := Sum[n^(n - 2) x^n/n!, {n, 1, nn}];
Map[Select[#, # > 0 &] &,
  Range[0, nn]! CoefficientList[
    Series[f[x + y x] + 1, {x, 0, nn}], {x, y}]] // Grid

E.g.f.: A(x + y*x) where A(x) is the e.g.f. for A000272.

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A185025 Triangular array read by rows: T(n,k) is the number of functions f:{1,2,...,n} -> {1,2,...,n} that have exactly k 2-cycles for n >= 0 and 0 <= k <= floor(n/2).

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A225213 Triangular array read by rows. T(n,k) is the number of cycles in the digraph representation of all functions f:{1,2,...,n}->{1,2,...,n} that have length k; 1<=k<=n.

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A345632 Sum of terms of even index in the binomial decomposition of n^(n-1).

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A345633 Sum of terms of odd index in the binomial decomposition of n^(n-1).

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A383036 The determinant of the matrix representing a totally anti-symmetric quasigroup of order 2*n+1.

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A249632 Triangular array read by rows. T(n,k) is the number of labeled trees with black and white nodes having exactly k black nodes, n>=0, 0<=k<=n.

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