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A246522 Number A(n,k) of endofunctions on [n] whose cycle lengths are divisors of k; square array A(n,k), n>=0, k>=0, read by antidiagonals. 11
1, 1, 0, 1, 1, 0, 1, 1, 3, 0, 1, 1, 4, 16, 0, 1, 1, 3, 25, 125, 0, 1, 1, 4, 18, 218, 1296, 0, 1, 1, 3, 25, 157, 2451, 16807, 0, 1, 1, 4, 16, 224, 1776, 33832, 262144, 0, 1, 1, 3, 27, 125, 2601, 24687, 554527, 4782969, 0, 1, 1, 4, 16, 250, 1320, 37072, 407464, 10535100, 100000000, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

LINKS

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

FORMULA

E.g.f. of column k: exp(Sum_{d|k} (-LambertW(-x))^d/d).

EXAMPLE

Square array A(n,k) begins:

  1,     1,     1,     1,     1,     1,     1, ...

  0,     1,     1,     1,     1,     1,     1, ...

  0,     3,     4,     3,     4,     3,     4, ...

  0,    16,    25,    18,    25,    16,    27, ...

  0,   125,   218,   157,   224,   125,   250, ...

  0,  1296,  2451,  1776,  2601,  1320,  2951, ...

  0, 16807, 33832, 24687, 37072, 17671, 42552, ...

MAPLE

with(numtheory):

egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):

A:= (n, k)-> n!*coeff(series(egf(k), x, n+1), x, n):

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

# second Maple program:

with(combinat):

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,

      add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1, k)*

      (i-1)!^j, j=0..`if`(irem(k, i)=0, n/i, 0))))

    end:

A:=(n, k)->add(b(j, min(k, j), k)*n^(n-j)*binomial(n-1, j-1), j=0..n):

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

MATHEMATICA

egf[k_] := Exp[Sum[(-ProductLog[-x])^d/d, {d, Divisors[k]}]]; A[1, 0] = 0; A[0, _] = 1; A[1, _] = 1; A[_, 0] = 0; A[n_, k_] := n!*SeriesCoefficient[egf[k], {x, 0, n}]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-Fran├žois Alcover, Dec 04 2014, translated from first Maple program *)

CROSSREFS

Columns k=0-10 give: A000007, A000272(n+1), A209319, A246523, A246524, A246525, A246526, A246527, A246528, A246529, A246530.

Main diagonal gives A246531.

Sequence in context: A029358 A088512 A094921 * A140166 A242782 A011256

Adjacent sequences:  A246519 A246520 A246521 * A246523 A246524 A246525

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Aug 28 2014

STATUS

approved

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Last modified June 24 17:11 EDT 2021. Contains 345417 sequences. (Running on oeis4.)