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A246071 Number of endofunctions f on [2n] satisfying f^n(i) = i for all i in [n].

%I A246071 #9 Jun 27 2022 07:51:06
%S A246071 1,2,50,1440,215760,11218000,8859219696,549669946784,797599992178688,
%T A246071 195297824029876992,225830701916170080000,33538442785393084937728,
%U A246071 478648537323384927696592896,26649057768458576467019134976,207869233649005397144301933676544
%N A246071 Number of endofunctions f on [2n] satisfying f^n(i) = i for all i in [n].
%H A246071 Alois P. Heinz, <a href="/A246071/b246071.txt">Table of n, a(n) for n = 0..100</a>
%F A246071 a(n) = A246070(2n,n).
%p A246071 with(numtheory): with(combinat): M:=multinomial:
%p A246071 b:= proc(n, k, p) local l, g; l, g:= sort([divisors(p)[]]),
%p A246071       proc(k, m, i, t) option remember; local d, j; d:= l[i];
%p A246071         `if`(i=1, n^m, add(M(k, k-(d-t)*j, (d-t)$j)/j!*
%p A246071          (d-1)!^j *M(m, m-t*j, t$j) *g(k-(d-t)*j, m-t*j,
%p A246071         `if`(d-t=1, [i-1, 0], [i, t+1])[]), j=0..min(k/(d-t),
%p A246071         `if`(t=0, [][], m/t))))
%p A246071       end; g(k, n-k, nops(l), 0)
%p A246071     end:
%p A246071 a:= n-> `if`(n=0, 1, b(2*n, n$2)):
%p A246071 seq(a(n), n=0..20);
%t A246071 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A246071 M = multinomial;
%t A246071 b[n_, k0_, p_] := Module[{l, g}, l = Divisors[p];
%t A246071    g[k_, m_, i_, t_] := g[k, m, i, t] = Module[{d, j}, d = l[[i]];
%t A246071    If[i == 1, If[m == 0, 1, n^m],  Sum[M[k, Join[{k - (d - t)*j},
%t A246071    Table[d - t, {j}]]]/j!*If[j == 0, 1, (d - 1)!^j]*M[m, Join[{m - t*j},
%t A246071    Array[t&, j]]]*g[k - (d - t)*j, m - t*j, Sequence @@
%t A246071    If[d - t == 1, {i - 1, 0}, {i, t + 1}]], {j, 0, Min[k/(d - t),
%t A246071    If[t == 0, {}, m/t]]}]]];
%t A246071    g[k0, n - k0, Length[l], 0]];
%t A246071 a[n_] := If[n == 0, 1, b[2*n, n, n]];
%t A246071 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jun 27 2022, after _Alois P. Heinz_ *)
%Y A246071 Cf. A246070.
%K A246071 nonn
%O A246071 0,2
%A A246071 _Alois P. Heinz_, Aug 12 2014