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A246531 Number of endofunctions on [n] whose cycle lengths are divisors of n. 2
1, 1, 4, 18, 224, 1320, 42552, 262864, 12232320, 169594560, 6117023600, 61920993024, 8022787347456, 56694391376896, 5193025319432160, 174746314698336000, 10338252997184749568, 121439552019384139776, 26096843176349347142208, 262144006402373705728000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..200

FORMULA

a(n) = n! * [x^n] exp(Sum_{d|n} (-LambertW(-x))^d/d).

a(n) = A246522(n,n).

MAPLE

with(numtheory):

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

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

seq(a(n), n=0..20);

# 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-> add(b(j$2, n)*n^(n-j)*binomial(n-1, j-1), j=0..n):

seq(a(n), n=0..20);

CROSSREFS

Main diagonal of A246522.

Sequence in context: A071173 A143993 A145660 * A278565 A214168 A214189

Adjacent sequences:  A246528 A246529 A246530 * A246532 A246533 A246534

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Aug 28 2014

STATUS

approved

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Last modified May 9 06:30 EDT 2021. Contains 343692 sequences. (Running on oeis4.)