A245958 -id:A245958 - cp's OEIS frontend

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

1, 1, 6, 141, 6184, 387545, 33404256, 3891981205, 592320594048, 113184611671473, 26327424526220800, 7302855260707822541, 2381136881374877847552, 901709366369630531857417, 392234247731566637785780224, 194028806625479344354551301125

Offset: 0

Alois P. Heinz, Aug 07 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..150

Cf. A048993, A239750, A239771, A245958.

Column k=3 of A245980.

with(combinat): M:=multinomial:
b:= proc(n, k) local l, g; l, g:= [1, 3],
      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-> add(b(n, j)*stirling2(n, j)*binomial(n, j)*j!, j=0..n):
seq(a(n), n=0..20);

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 *)

a(n) = Sum_{k=0..n} C(n,k) * A048993(n,k) * k! * A245958(n,k).

A245959 Number of endofunctions f on [2n] satisfying f^3(i) = i for all i in [n].

Original entry on oeis.org

1, 2, 36, 1440, 84624, 7675200, 962250624, 151851992544, 30421572307200, 7430515709340672, 2142144445293849600, 727442024443449689088, 285148327160858698469376, 127152465871110917459189760, 64226006581334387301393186816, 36322574197169989225245335040000

Offset: 0

Alois P. Heinz, Aug 08 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..150

Cf. A245958.

Column k=3 of A246070.

with(combinat): M:=multinomial:
T:= proc(n, k) local l, g; l, g:= [1, 3],
      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-> T(2*n, n):
seq(a(n), n=0..20);

M[n_, m_, k_List] := n!/Times @@ (Join[{m}, k]!);
T[0, 0] = 1; T[n_, k_] := T[n, k] = Module[{l = {1, 3}, g}, g[k0_, m_, {i_, t_}] := g[k0, m, i, t]; g[k0_, m_, i_, t_] := g[k0, m, i, t] = Module[ {d}, d = l[[i]]; If[i == 1, n^m, Sum[M[k0, k0 - (d - t)*j, Table[(d - t), {j}]]/j!*(d - 1)!^j*M[m, m - t*j, Table[t, {j}]]*g[k0 - (d - t)*j, m - t*j, If[d - t == 1, {i - 1, 0}, {i, t + 1}]], {j, 0, Min[k0/(d - t), If[t == 0, Infinity, m/t]]}]]]; g[k, n - k, Length[l], 0]];
a[n_] := T[2 n, n];
a /@ Range[0, 20] (* Jean-François Alcover, Oct 03 2019, after Alois P. Heinz *)

a(n) = A245958(2n,n).

cp's OEIS Frontend

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

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