A224246 The number of n-permutations that have a unique smallest cycle and this cycle contains the element 1.
1, 1, 3, 8, 41, 194, 1309, 9022, 79057, 689588, 7462601, 80632826, 1021071193, 13120783948, 192752054377, 2848878770774, 47617784530529, 800500650553472, 14910497765819137, 281133366288649138, 5803224036600349801, 120681837753825004796, 2734647516979262677673, 62424209302423879016558, 1535507329367939907583057
Offset: 1
Keywords
Examples
a(4) = 8 because we have the permutations of {1,2,3,4} in cycle notation:
{{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}}.
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..450
Programs
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Maple
b:= proc(n, t) option remember; `if`(n=0, 1, add((i-1)!* binomial(n-1, i-1)*b(n-i, `if`(t=1, i+1, t)), i=t..n)) end: a:= n-> b(n, 1): seq(a(n), n=1..30); # Alois P. Heinz, Sep 07 2020 -
Mathematica
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]
Formula
E.g.f.: Sum_{k>=1} Integral_((x^(k-1)/(k-1))*exp(-Sum_{i=1..k} x^i/i)/(1-x) dx).