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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.

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