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.

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, 4, 12, 110, 196, 46656, 1, 1, 1, 3, 19, 73, 751, 1057, 823543, 1, 1, 1, 4, 10, 116, 556, 5902, 6322, 16777216, 1, 1, 1, 3, 21, 41, 901, 4737, 52165, 41393, 387420489, 1

Offset: 0

Author

Alois P. Heinz, Jul 24 2014

Keywords

• nonn
• tabl

Examples

			Square array A(n,k) begins:
  1,     1,    1,    1,    1,    1,    1, ...
  1,     1,    1,    1,    1,    1,    1, ...
  1,     4,    3,    4,    3,    4,    3, ...
  1,    27,   10,   19,   12,   19,   10, ...
  1,   256,   41,  110,   73,  116,   41, ...
  1,  3125,  196,  751,  556,  901,  220, ...
  1, 46656, 1057, 5902, 4737, 8422, 1921, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened

Crossrefs

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

Main diagonal gives A245507.

Programs

Maple

with(numtheory):
A:= (n, k)-> `if`(k=0, 1, `if`(k=1, n^n, n! *coeff(series(
    exp(add((x*exp(x))^d/d, d=divisors(k-1))), x, n+1), x, n))):
seq(seq(A(n, d-n), n=0..d), d=0..12);

Mathematica

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 *)

Formula

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.