A245910 Number A(n,k) of pairs of endofunctions f, g on [n] satisfying f(g^k(i)) = f(i) for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 16, 1, 1, 10, 729, 1, 1, 12, 159, 65536, 1, 1, 10, 249, 3496, 9765625, 1, 1, 12, 207, 7744, 98345, 2176782336, 1, 1, 10, 249, 6856, 326745, 3373056, 678223072849, 1, 1, 12, 159, 9184, 302345, 17773056, 136535455, 281474976710656

Offset

0,6

Links

Alois P. Heinz, Antidiagonals n = 0..100, flattened

Example

Square array A(n,k) begins:
0 :        1,     1,      1,      1,      1,      1, ...
1 :        1,     1,      1,      1,      1,      1, ...
2 :       16,    10,     12,     10,     12,     10, ...
3 :      729,   159,    249,    207,    249,    159, ...
4 :    65536,  3496,   7744,   6856,   9184,   3496, ...
5 :  9765625, 98345, 326745, 302345, 488745, 173225, ...

Maple

with(combinat):
b:= proc(n, i, k) option remember; unapply(`if`(n=0 or i=1, x^n,
      expand(add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
      x^(igcd(i, k)*j)*b(n-i*j, i-1, k)(x), j=0..n/i))), x)
    end:
A:= (n, k)-> `if`(k=0, n^(2*n), add(binomial(n-1, j-1)*n^(n-j)*
              b(j$2, k)(n), j=0..n)):
seq(seq(A(n, d-n), n=0..d), d=0..10);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_, k_] := b[n, i, k] = Function[{x}, If[n == 0 || i == 1, x^n, Expand[Sum[(i-1)!^j*multinomial[n, Join[{ n-i*j}, Array[i&, j]]]/j!*x^(GCD[i, k]*j)*b[n-i*j, i-1, k][x], {j, 0, n/i}]]]]; A[0, _] = 1; A[n_, k_] := If[k == 0, n^(2n), Sum[Binomial[n-1, j-1]*n^(n-j)* b[j, j, k][n], {j, 0, n}]]; Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 04 2015, after Alois P. Heinz *)

Crossrefs

Columns k=0-10 give: A062206, A239761, A239777, A245912, A245913, A245914, A245915, A245916, A245917, A245918, A245919.

Main diagonal gives A245911.

Cf. A061356, A245980.

Sequence in context: A040257 A245980 A040256 * A133824 A154228 A141697

Adjacent sequences: A245907 A245908 A245909 * A245911 A245912 A245913

Keyword

nonn,tabl

Author

Alois P. Heinz, Aug 06 2014

Status

approved