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A246070 Number A(n,k) of endofunctions f on [2n] satisfying f^k(i) = i for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals. 7
1, 1, 4, 1, 2, 256, 1, 3, 16, 46656, 1, 2, 50, 216, 16777216, 1, 3, 36, 1626, 4096, 10000000000, 1, 2, 56, 1440, 83736, 100000, 8916100448256, 1, 3, 16, 2688, 84624, 6026120, 2985984, 11112006825558016, 1, 2, 70, 720, 215760, 7675200, 571350096, 105413504, 18446744073709551616 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Antidiagonals n = 0..70, flattened

EXAMPLE

Square array A(n,k) begins:

0 : 1, 1, 1, 1, 1, 1, ...

1 : 4, 2, 3, 2, 3, 2, ...

2 : 256, 16, 50, 36, 56, 16, ...

3 : 46656, 216, 1626, 1440, 2688, 720, ...

4 : 16777216, 4096, 83736, 84624, 215760, 94816, ...

5 : 10000000000, 100000, 6026120, 7675200, 24899120, 11218000, ...

MAPLE

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, k)-> `if`(k=0, (2*n)^(2*n), b(2*n, n, k)):

seq(seq(A(n, d-n), n=0..d), d=0..10);

MATHEMATICA

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_, k_] := If[k == 0, If[n == 0, 1, (2n)^(2n)], b[2*n, n, k]];

Table[A[n, d - n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 27 2016, after Alois P. Heinz, updated Jan 01 2021 *)

CROSSREFS

Columns k=0-3 give: A085534, A062971, A245141, A245959.

Main diagonal gives A246071.

Cf. A246072 (the same for permutations).

Sequence in context: A280284 A004161 A303141 * A202778 A025016 A355499

Adjacent sequences: A246067 A246068 A246069 * A246071 A246072 A246073

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Aug 12 2014

STATUS

approved

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Last modified November 29 18:13 EST 2022. Contains 358431 sequences. (Running on oeis4.)