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A245501 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.

%I A245501 #20 Feb 24 2019 11:29:51
%S A245501 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 A245501 4,12,110,196,46656,1,1,1,3,19,73,751,1057,823543,1,1,1,4,10,116,556,
%U A245501 5902,6322,16777216,1,1,1,3,21,41,901,4737,52165,41393,387420489,1
%N A245501 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 A245501 Alois P. Heinz, <a href="/A245501/b245501.txt">Antidiagonals n = 0..140, flattened</a>
%F A245501 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 A245501 Square array A(n,k) begins:
%e A245501   1,     1,    1,    1,    1,    1,    1, ...
%e A245501   1,     1,    1,    1,    1,    1,    1, ...
%e A245501   1,     4,    3,    4,    3,    4,    3, ...
%e A245501   1,    27,   10,   19,   12,   19,   10, ...
%e A245501   1,   256,   41,  110,   73,  116,   41, ...
%e A245501   1,  3125,  196,  751,  556,  901,  220, ...
%e A245501   1, 46656, 1057, 5902, 4737, 8422, 1921, ...
%p A245501 with(numtheory):
%p A245501 A:= (n, k)-> `if`(k=0, 1, `if`(k=1, n^n, n! *coeff(series(
%p A245501     exp(add((x*exp(x))^d/d, d=divisors(k-1))), x, n+1), x, n))):
%p A245501 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A245501 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 A245501 Column k=0-10 give: A000012, A000312, A000248, A060905, A060906, A060907, A245502, A245503, A245504, A245505, A245506.
%Y A245501 Main diagonal gives A245507.
%K A245501 nonn,tabl
%O A245501 0,9
%A A245501 _Alois P. Heinz_, Jul 24 2014