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A246071
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Number of endofunctions f on [2n] satisfying f^n(i) = i for all i in [n].
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2
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1, 2, 50, 1440, 215760, 11218000, 8859219696, 549669946784, 797599992178688, 195297824029876992, 225830701916170080000, 33538442785393084937728, 478648537323384927696592896, 26649057768458576467019134976, 207869233649005397144301933676544
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OFFSET
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0,2
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LINKS
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FORMULA
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MAPLE
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with(numtheory): with(combinat): M:=multinomial:
b:= proc(n, k, p) local l, g; l, g:= sort([divisors(p)[]]),
proc(k, m, i, t) option remember; local d, j; d:= l[i];
`if`(i=1, n^m, add(M(k, k-(d-t)*j, (d-t)$j)/j!*
(d-1)!^j *M(m, m-t*j, t$j) *g(k-(d-t)*j, m-t*j,
`if`(d-t=1, [i-1, 0], [i, t+1])[]), j=0..min(k/(d-t),
`if`(t=0, [][], m/t))))
end; g(k, n-k, nops(l), 0)
end:
a:= n-> `if`(n=0, 1, b(2*n, n$2)):
seq(a(n), n=0..20);
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MATHEMATICA
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multinomial[n_, k_List] := n!/Times @@ (k!);
M = multinomial;
b[n_, k0_, p_] := Module[{l, g}, l = Divisors[p];
g[k_, m_, i_, t_] := g[k, m, i, t] = Module[{d, j}, d = l[[i]];
If[i == 1, If[m == 0, 1, n^m], Sum[M[k, Join[{k - (d - t)*j},
Table[d - t, {j}]]]/j!*If[j == 0, 1, (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, {}, m/t]]}]]];
g[k0, n - k0, Length[l], 0]];
a[n_] := If[n == 0, 1, b[2*n, n, n]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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