A246524 Number of endofunctions on [n] whose cycle lengths are divisors of 4.

1, 1, 4, 25, 224, 2601, 37072, 626137, 12227280, 271086625, 6727858496, 184818121929, 5568152828416, 182575550335465, 6473161538599680, 246781048203043561, 10067677495565927168, 437653901985319521153, 20197310874805488471040, 986221173076368356013625

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..350

Formula

E.g.f.: exp(Sum_{d|4} (-LambertW(-x))^d/d).

Maple

with(numtheory):
egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):
a:= n-> n!*coeff(series(egf(4), x, n+1), x, n):
seq(a(n), n=0..25);
# second Maple program:
with(combinat):
b:= proc(n, i) 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)*
      (i-1)!^j, j=0..`if`(irem(4, i)=0, n/i, 0))))
    end:
a:= n-> add(b(j, min(4, j))*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);

Crossrefs

Column k=4 of A246522.

Sequence in context: A246530 A268163 A050386 * A246528 A246951 A302587

Adjacent sequences: A246521 A246522 A246523 * A246525 A246526 A246527

Keyword

nonn

Author

Alois P. Heinz, Aug 28 2014

Status

approved