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A246527 Number of endofunctions on [n] whose cycle lengths are divisors of 7.

%I A246527 #5 Aug 28 2014 20:18:25
%S A246527 1,1,3,16,125,1296,16807,262864,4829049,102073600,2441582891,
%T A246527 65201946624,1922453391157,62009850843136,2171369477933775,
%U A246527 82007515430081536,3322113623606686193,143662773881554108416,6604529623711334804179,321608928954695680000000
%N A246527 Number of endofunctions on [n] whose cycle lengths are divisors of 7.
%H A246527 Alois P. Heinz, <a href="/A246527/b246527.txt">Table of n, a(n) for n = 0..350</a>
%F A246527 E.g.f.: exp(Sum_{d|7} (-LambertW(-x))^d/d).
%p A246527 with(numtheory):
%p A246527 egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):
%p A246527 a:= n-> n!*coeff(series(egf(7), x, n+1), x, n):
%p A246527 seq(a(n), n=0..25);
%p A246527 # second Maple program:
%p A246527 with(combinat):
%p A246527 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A246527       add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)*
%p A246527       (i-1)!^j, j=0..`if`(irem(7, i)=0, n/i, 0))))
%p A246527     end:
%p A246527 a:= n-> add(b(j, min(7, j))*n^(n-j)*binomial(n-1, j-1), j=0..n):
%p A246527 seq(a(n), n=0..25);
%Y A246527 Column k=7 of A246522.
%K A246527 nonn
%O A246527 0,3
%A A246527 _Alois P. Heinz_, Aug 28 2014