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Number of pairs of endofunctions f, g on [n] satisfying f(g^4(i)) = f(i) for all i in [n].
2

%I #4 Aug 06 2014 15:15:17

%S 1,1,12,249,9184,488745,35463456,3212948809,369653885952,

%T 52089606360081,8863922597593600,1752215974571247801,

%U 402913941534323970048,106177876504463493003001,31939024924944619647750144,10756222724503803551432639625,4050020577581980281160989147136

%N Number of pairs of endofunctions f, g on [n] satisfying f(g^4(i)) = f(i) for all i in [n].

%H Alois P. Heinz, <a href="/A245913/b245913.txt">Table of n, a(n) for n = 0..150</a>

%p with(combinat):

%p T:= proc(n, j) option remember; binomial(n-1, j-1)*n^(n-j) end:

%p b:= proc(n, i) option remember; `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, 4)*j)*b(n-i*j, i-1), j=0..n/i)))

%p end:

%p a:= n-> add((p-> add(n^i*T(n, j)* coeff(p, x, i),

%p i=0..degree(p)))(b(j$2)), j=0..n):

%p seq(a(n), n=0..20);

%Y Column k=4 of A245910.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Aug 06 2014