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A224246 The number of n-permutations that have a unique smallest cycle and this cycle contains the element 1.

%I A224246 #15 Jan 05 2021 03:02:12
%S A224246 1,1,3,8,41,194,1309,9022,79057,689588,7462601,80632826,1021071193,
%T A224246 13120783948,192752054377,2848878770774,47617784530529,
%U A224246 800500650553472,14910497765819137,281133366288649138,5803224036600349801,120681837753825004796,2734647516979262677673,62424209302423879016558,1535507329367939907583057
%N A224246 The number of n-permutations that have a unique smallest cycle and this cycle contains the element 1.
%H A224246 Alois P. Heinz, <a href="/A224246/b224246.txt">Table of n, a(n) for n = 1..450</a>
%F A224246 E.g.f.: Sum_{k>=1} Integral_((x^(k-1)/(k-1))*exp(-Sum_{i=1..k} x^i/i)/(1-x) dx).
%e A224246 a(4) = 8 because we have the permutations of {1,2,3,4} in cycle notation:
%e A224246 {{1}, {3,4,2}}, {{1}, {4,3,2}}, {{2,3,4,1}}, {{2,4,3,1}}, {{3,4,2,1}}, {{3,2,4,1}}, {{4,3,2,1}}, {{4,2,3,1}}.
%p A224246 b:= proc(n, t) option remember; `if`(n=0, 1, add((i-1)!*
%p A224246       binomial(n-1, i-1)*b(n-i, `if`(t=1, i+1, t)), i=t..n))
%p A224246     end:
%p A224246 a:= n-> b(n, 1):
%p A224246 seq(a(n), n=1..30);  # _Alois P. Heinz_, Sep 07 2020
%t A224246 nn=20; Drop[Range[0,nn]! CoefficientList[Series[Sum[Integrate[x^(k-1) Exp[-Sum[x^i/i,{i,1,k}]]/(1-x),x], {k,1,nn}], {x,0,nn}], x],1]
%Y A224246 Cf. A224245, A224244, A224219.
%K A224246 nonn
%O A224246 1,3
%A A224246 _Geoffrey Critzer_, Apr 01 2013