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

0,9

Links

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

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.

Example

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, ...

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 *)

Crossrefs

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

Main diagonal gives A245507.

Sequence in context: A010328 A378819 A332847 * A247026 A193512 A366524

Adjacent sequences: A245498 A245499 A245500 * A245502 A245503 A245504

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 24 2014

Status

approved