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%I #14 Sep 07 2020 14:22:05
%S 1,1,5,14,89,474,3499,27040,253161,2426300,27596051,323960856,
%T 4277055925,59041067344,898062119655,14172430400864,243919993681649,
%U 4347177953716080,83224487266425811,1653277176082392040,34961357216796300381,763702067489722288136
%N Number of n-permutations in which there is a unique smallest cycle.
%C In other words, if the smallest cycle in the n-permutation has length k then no other cycle in the permutation has length k.
%H Alois P. Heinz, <a href="/A224245/b224245.txt">Table of n, a(n) for n = 1..449</a>
%F E.g.f.: Sum_{k>=1} x^k/k * exp(-Sum_{i=1..k}x^i/i)/(1-x).
%e a(4) = 14 because we have 14 such permutations of {1,2,3,4} shown in cycle notation: {{1}, {3,4,2}}, {{1}, {4,3,2}}, {{2,3,1}, {4}}, {{2,3,4,1}}, {{2,4,3,1}}, {{2,4,1}, {3}}, {{3,2,1}, {4}}, {{3,4,2,1}}, {{3,4,1}, {2}}, {{3,2,4,1}}, {{4,3,2,1}}, {{4,2,1}, {3}}, {{4,3,1}, {2}}, {{4,2,3,1}}.
%p with(combinat):
%p b:= proc(n, i) option remember;
%p `if`(i<1, 0, `if`(n=i, (i-1)!, 0) +add(b(n-i*j, i-1)*
%p multinomial(n, n-i*j, i$j)/j!*(i-1)!^j, j=0..(n-1)/i))
%p end:
%p a:= n-> b(n$2):
%p seq(a(n), n=1..25); # _Alois P. Heinz_, Sep 07 2020
%t nn=20; Drop[Range[0,nn]! CoefficientList[Series[Sum[x^k/k Exp[-Sum[x^i/i, {i,1,k}]]/(1-x), {k,1,nn}], {x,0,nn}], x], 1]
%Y Cf. A224219, A224244.
%K nonn
%O 1,3
%A _Geoffrey Critzer_, Apr 01 2013
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