%I #20 Feb 24 2019 11:29:51

%S 1,1,1,1,1,1,1,1,4,1,1,1,3,27,1,1,1,4,10,256,1,1,1,3,19,41,3125,1,1,1,

%T 4,12,110,196,46656,1,1,1,3,19,73,751,1057,823543,1,1,1,4,10,116,556,

%U 5902,6322,16777216,1,1,1,3,21,41,901,4737,52165,41393,387420489,1

%N Number A(n,k) of endofunctions f on [n] such that f^k(i) = f(i) for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%H Alois P. Heinz, <a href="/A245501/b245501.txt">Antidiagonals n = 0..140, flattened</a>

%F A(n,k) = n! * [x^n] exp(Sum_{d|(k-1)} (x*exp(x))^d/d) for k>1, A(n,0)=1, A(n,1)=n^n.

%e Square array A(n,k) begins:

%e 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 4, 3, 4, 3, 4, 3, ...

%e 1, 27, 10, 19, 12, 19, 10, ...

%e 1, 256, 41, 110, 73, 116, 41, ...

%e 1, 3125, 196, 751, 556, 901, 220, ...

%e 1, 46656, 1057, 5902, 4737, 8422, 1921, ...

%p with(numtheory):

%p A:= (n, k)-> `if`(k=0, 1, `if`(k=1, n^n, n! *coeff(series(

%p exp(add((x*exp(x))^d/d, d=divisors(k-1))), x, n+1), x, n))):

%p seq(seq(A(n, d-n), n=0..d), d=0..12);

%t A[0, 1] = 1; A[n_, k_] := If[k==0, 1, If[k==1, n^n, n!*SeriesCoefficient[ Exp[ DivisorSum[k-1, (x*Exp[x])^#/#&]], {x, 0, n}]]]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Mar 20 2017, translated from Maple *)

%Y Column k=0-10 give: A000012, A000312, A000248, A060905, A060906, A060907, A245502, A245503, A245504, A245505, A245506.

%Y Main diagonal gives A245507.

%K nonn,tabl

%O 0,9

%A _Alois P. Heinz_, Jul 24 2014