A245692 -id:A245692 - cp's OEIS frontend

A076728 a(n) = (n-1)^2 * n^(n-2).

0, 1, 12, 144, 2000, 32400, 605052, 12845056, 306110016, 8100000000, 235794769100, 7492001071104, 258071096741328, 9581271191425024, 381454233398437500, 16212958658533785600, 732780301186512843008, 35096024486915738763264, 1775645341922275908244236

Offset: 1

Benoit Cloitre, Oct 25 2002

nonn

Smallest integer value of the form 1/z(k,n) where z(k,x)=x/(x-1)^2 -sum(i=1,k,i/x^i).
For any x>1 lim k -> infinity z(k,x)=0. More generally if p is an integer >=2, 1/z(u(k),p) is an integer for any k>=2 where u(k)=(p-1)^2*p^((p^k-(p-1)*k-p)/(p-1)). u(k) can also be written : u(k)=(p-1)^2 *p^(1+p+p^2+...+p^(k-2)).
For n>=2, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,...,n} such that for fixed, different x_1, x_2 in {1,2,...,n} and fixed y_1, y_2 in {1,2,...,n} we have f(x_1)<>y_1 and f(x_2)<> y_2. - Milan Janjic, May 10 2007
a(n+1) = Sum_{k=0...n} binomial(n,k)*n^k*k, which enumerates the total number of elements in the domain of definition over all partial functions on n labeled objects. - Geoffrey Critzer, Feb 08 2012
Also, the number of possible negation tables in the n-valued logics (cf. A262458 and A262459). - Max Alekseyev, Sep 23 2015

Alois P. Heinz, Table of n, a(n) for n = 1..386
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Column k=0 of A245692.

Table[Sum[Binomial[n,k] n^k k, {k,0,n}], {n,1,20}] (* Geoffrey Critzer, Feb 08 2012 *)

PARI
```
a(n) = (n-1)^2*n^(n-2)
```

a(1)=0 prepended by Max Alekseyev, Sep 23 2015

Some terms corrected by Alois P. Heinz, May 22 2016

A245348 Number T(n,k) of endofunctions f on [n] that are self-inverse on [k]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 4, 3, 2, 27, 15, 8, 4, 256, 112, 50, 22, 10, 3125, 1125, 430, 166, 66, 26, 46656, 14256, 4752, 1626, 576, 206, 76, 823543, 218491, 64484, 19768, 6310, 2054, 688, 232, 16777216, 3932160, 1040384, 288512, 83736, 24952, 7660, 2388, 764

Offset: 0

Alois P. Heinz, Jul 18 2014

T(n,k) counts endofunctions f:{1,...,n}-> {1,...,n} with f(f(i))=i for all i in {1,...,k}.

			T(3,1) = 15: (1,1,1), (2,1,1), (3,1,1), (1,2,1), (3,2,1), (1,3,1), (3,3,1), (1,1,2), (2,1,2), (1,2,2), (1,3,2), (1,1,3), (2,1,3), (1,2,3), (1,3,3).
T(3,2) = 8: (2,1,1), (1,2,1), (3,2,1), (2,1,2), (1,2,2), (1,3,2), (2,1,3), (1,2,3).
T(3,3) = 4: (3,2,1), (1,3,2), (2,1,3), (1,2,3).
Triangle T(n,k) begins:
0 :       1;
1 :       1,      1;
2 :       4,      3,     2;
3 :      27,     15,     8,     4;
4 :     256,    112,    50,    22,   10;
5 :    3125,   1125,   430,   166,   66,   26;
6 :   46656,  14256,  4752,  1626,  576,  206,  76;
7 :  823543, 218491, 64484, 19768, 6310, 2054, 688, 232;
     ...

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

Columns k=0-1 give: A000312, A089945(n-1) for n>0.
Main diagonal gives A000085.
T(2n,n) gives A245141.
Cf. A239771, A245692.

g:= proc(n) g(n):= `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end:
T:= (n, k)-> add(binomial(n-k, i)*binomial(k, i)*i!*
             g(k-i)*n^(n-k-i), i=0..min(k, n-k)):
seq(seq(T(n,k), k=0..n), n=0..10);

g[n_] := g[n] = If[n<2, 1, g[n-1] + (n-1)*g[n-2]]; T[0, 0] = 1; T[n_, k_] := Sum[Binomial[n-k, i]*Binomial[k, i]*i!*g[k-i]*n^(n-k-i), {i, 0, Min[k, n-k]}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 19 2017, translated from Maple *)

T(n,k) = Sum_{i=0..min(k,n-k)} C(n-k,i)*C(k,i)*i!*A000085(k-i)*n^(n-k-i).

A245693 Number T(n,k) of permutations on [n] that are self-inverse on [k] but not on [k+1]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 2, 0, 0, 4, 12, 2, 0, 0, 10, 72, 18, 4, 0, 0, 26, 480, 120, 36, 8, 0, 0, 76, 3600, 840, 264, 84, 20, 0, 0, 232, 30240, 6480, 1920, 648, 216, 52, 0, 0, 764, 282240, 55440, 15120, 4920, 1776, 612, 152, 0, 0, 2620, 2903040, 524160, 131040, 39600, 13920, 5232, 1848, 464, 0, 0, 9496

Offset: 0

Alois P. Heinz, Jul 29 2014

T(n,k) counts permutations p:{1,...,n}-> {1,...,n} with p(p(i))=i for all i in {1,...,k} and p(p(k+1))<>k+1 if k

			Triangle T(n,k) begins:
0 :      1;
1 :      0,    1;
2 :      0,    0,    2;
3 :      2,    0,    0,   4;
4 :     12,    2,    0,   0,  10;
5 :     72,   18,    4,   0,   0, 26;
6 :    480,  120,   36,   8,   0,  0, 76;
7 :   3600,  840,  264,  84,  20,  0,  0, 232;
8 :  30240, 6480, 1920, 648, 216, 52,  0,   0, 764;

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

Column k=0 give A062119(n-1) for n>1.
Row sums give A000142.
Main diagonal gives A000085.
Cf. A245692 (the same for endofunctions).

g:= proc(n) g(n):= `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end:
H:= (n, k)-> add(binomial(n-k, i)*binomial(k, i)*i!*
             g(k-i)*(n-k-i)!, i=0..min(k, n-k)):
T:= (n, k)-> H(n, k) -H(n, k+1):
seq(seq(T(n, k), k=0..n), n=0..10);

g[n_] := g[n] = If[n < 2, 1, g[n - 1] + (n - 1)*g[n - 2]];
H[n_, k_] := Sum[Binomial[n - k, i]*Binomial[k, i]*i!*
     g[k - i]*(n - k - i)!, {i, 0, Min[k, n - k]}];
T[n_, k_] := H[n, k] - H[n, k + 1];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 10 2021, after Alois P. Heinz *)

T(n,k) = H(n,k) - H(n,k+1) with H(n,k) = Sum_{i=0..min(k,n-k)} C(n-k,i) * C(k,i) * i! * A000085(k-i) * (n-k-i)!.

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A076728 a(n) = (n-1)^2 * n^(n-2).

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A245348 Number T(n,k) of endofunctions f on [n] that are self-inverse on [k]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A245693 Number T(n,k) of permutations on [n] that are self-inverse on [k] but not on [k+1]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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