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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. 11
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A321592 A031278 A010328 * A247026 A193512 A276330

Adjacent sequences:  A245498 A245499 A245500 * A245502 A245503 A245504

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jul 24 2014

STATUS

approved

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Last modified January 23 22:36 EST 2020. Contains 331177 sequences. (Running on oeis4.)