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A241015 Number of pairs of endofunctions f, g on [n] satisfying g(g(g(f(i)))) = f(i) for all i in [n].

%I A241015 #15 Mar 13 2017 04:22:11
%S A241015 1,1,6,141,6184,387545,33404256,3891981205,592320594048,
%T A241015 113184611671473,26327424526220800,7302855260707822541,
%U A241015 2381136881374877847552,901709366369630531857417,392234247731566637785780224,194028806625479344354551301125
%N A241015 Number of pairs of endofunctions f, g on [n] satisfying g(g(g(f(i)))) = f(i) for all i in [n].
%H A241015 Alois P. Heinz, <a href="/A241015/b241015.txt">Table of n, a(n) for n = 0..150</a>
%F A241015 a(n) = Sum_{k=0..n} C(n,k) * A048993(n,k) * k! * A245958(n,k).
%p A241015 with(combinat): M:=multinomial:
%p A241015 b:= proc(n, k) local l, g; l, g:= [1, 3],
%p A241015       proc(k, m, i, t) option remember; local d, j; d:= l[i];
%p A241015         `if`(i=1, n^m, add(M(k, k-(d-t)*j, (d-t)$j)/j!*
%p A241015          (d-1)!^j *M(m, m-t*j, t$j) *g(k-(d-t)*j, m-t*j,
%p A241015         `if`(d-t=1, [i-1, 0], [i, t+1])[]), j=0..min(k/(d-t),
%p A241015         `if`(t=0, [][], m/t))))
%p A241015       end; g(k, n-k, nops(l), 0)
%p A241015     end:
%p A241015 a:= n-> add(b(n, j)*stirling2(n, j)*binomial(n, j)*j!, j=0..n):
%p A241015 seq(a(n), n=0..20);
%t A241015 multinomial[n_, k_] := n!/Times @@ (k!); M = multinomial; b[n_, k0_] := Module[{l, g}, l = {1, 3}; g[k_, m_, i_, t_] := g[k, m, i, t] = Module[{d, j}, d = l[[i]]; If[i==1, n^m, Sum[M[k, Join[{k-(d-t)*j}, Array[d-t&, j]]]/j!*(d-1)!^j *M[m, Join[{m-t*j}, Array[t&, j]]]*g[k-(d-t)*j, m-t*j, Sequence @@ If[d-t==1, {i-1, 0}, {i, t+1}]], {j, 0, Min[k/(d-t), If[t==0, Infinity, m/t]]}]]]; g[k0, n-k0, Length[l], 0]]; a[0] = 1; a[n_] := Sum[b[n, j]*StirlingS2[n, j]*Binomial[n, j]*j!, {j, 0, n}]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Mar 13 2017, translated from Maple *)
%Y A241015 Cf. A048993, A239750, A239771, A245958.
%Y A241015 Column k=3 of A245980.
%K A241015 nonn
%O A241015 0,3
%A A241015 _Alois P. Heinz_, Aug 07 2014