%I #18 Feb 04 2015 09:00:01
%S 1,1,1,1,1,16,1,1,10,729,1,1,12,159,65536,1,1,10,249,3496,9765625,1,1,
%T 12,207,7744,98345,2176782336,1,1,10,249,6856,326745,3373056,
%U 678223072849,1,1,12,159,9184,302345,17773056,136535455,281474976710656
%N 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.
%H Alois P. Heinz, <a href="/A245910/b245910.txt">Antidiagonals n = 0..100, flattened</a>
%e Square array A(n,k) begins:
%e 0 : 1, 1, 1, 1, 1, 1, ...
%e 1 : 1, 1, 1, 1, 1, 1, ...
%e 2 : 16, 10, 12, 10, 12, 10, ...
%e 3 : 729, 159, 249, 207, 249, 159, ...
%e 4 : 65536, 3496, 7744, 6856, 9184, 3496, ...
%e 5 : 9765625, 98345, 326745, 302345, 488745, 173225, ...
%p with(combinat):
%p b:= proc(n, i, k) option remember; unapply(`if`(n=0 or i=1, x^n,
%p expand(add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
%p x^(igcd(i, k)*j)*b(n-i*j, i-1, k)(x), j=0..n/i))), x)
%p end:
%p A:= (n, k)-> `if`(k=0, n^(2*n), add(binomial(n-1, j-1)*n^(n-j)*
%p b(j$2, k)(n), j=0..n)):
%p seq(seq(A(n, d-n), n=0..d), d=0..10);
%t 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_ *)
%Y Columns k=0-10 give: A062206, A239761, A239777, A245912, A245913, A245914, A245915, A245916, A245917, A245918, A245919.
%Y Main diagonal gives A245911.
%Y Cf. A061356, A245980.
%K nonn,tabl
%O 0,6
%A _Alois P. Heinz_, Aug 06 2014