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A261430 Number A(n,k) of permutations p of [n] without fixed points such that p^k = Id; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A261430 #10 Oct 27 2018 16:44:39
%S A261430 1,1,0,1,0,0,1,0,0,0,1,0,1,0,0,1,0,0,0,0,0,1,0,1,2,3,0,0,1,0,0,0,0,0,
%T A261430 0,0,1,0,1,0,9,0,15,0,0,1,0,0,2,0,0,40,0,0,0,1,0,1,0,3,24,105,0,105,0,
%U A261430 0,1,0,0,0,0,20,0,0,0,0,0,0,1,0,1,2,9,0,175,0,2625,2240,945,0,0
%N A261430 Number A(n,k) of permutations p of [n] without fixed points such that p^k = Id; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A261430 Alois P. Heinz, <a href="/A261430/b261430.txt">Antidiagonals n = 0..140, flattened</a>
%F A261430 E.g.f. of column k: exp(Sum_{d|k, d>1} x^d/d).
%e A261430 Square array A(n,k) begins:
%e A261430   1, 1,   1,  1,    1,  1,    1,   1,    1, ...
%e A261430   0, 0,   0,  0,    0,  0,    0,   0,    0, ...
%e A261430   0, 0,   1,  0,    1,  0,    1,   0,    1, ...
%e A261430   0, 0,   0,  2,    0,  0,    2,   0,    0, ...
%e A261430   0, 0,   3,  0,    9,  0,    3,   0,    9, ...
%e A261430   0, 0,   0,  0,    0, 24,   20,   0,    0, ...
%e A261430   0, 0,  15, 40,  105,  0,  175,   0,  105, ...
%e A261430   0, 0,   0,  0,    0,  0,  210, 720,    0, ...
%e A261430   0, 0, 105,  0, 2625,  0, 4585,   0, 7665, ...
%p A261430 with(numtheory):
%p A261430 A:= proc(n, k) option remember; `if`(n<0, 0, `if`(n=0, 1,
%p A261430       add(mul(n-i, i=1..j-1)*A(n-j, k), j=divisors(k) minus {1})))
%p A261430     end:
%p A261430 seq(seq(A(n, d-n), n=0..d), d=0..14);
%t A261430 A[0, 0] = A[0, 1] = 1; A[_, 0|1] = 0; A[n_, k_] := A[n, k] = If[n < 0, 0, If[n == 0, 1, Sum[Product[n - i, {i, 1, j - 1}]*A[n - j, k], {j, Rest @ Divisors[k]}]]]; Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Jan 21 2017, after _Alois P. Heinz_ *)
%Y A261430 Columns k=0+1,2-10 give: A000007, A001147, A052502, A052503, A052504, A261317, A261427, A261428, A261429, A261381.
%Y A261430 Main diagonal gives A261431.
%Y A261430 Cf. A008307.
%K A261430 nonn,tabl
%O A261430 0,25
%A A261430 _Alois P. Heinz_, Aug 18 2015