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

1,3

Links

Alois P. Heinz, Table of n, a(n) for n = 1..450

Formula

E.g.f.: Sum_{k>=1} Integral_((x^(k-1)/(k-1))*exp(-Sum_{i=1..k} x^i/i)/(1-x) dx).

Example

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}}.

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]

Crossrefs

Cf. A224245, A224244, A224219.

Sequence in context: A152394 A168468 A330527 * A128322 A337758 A038048

Adjacent sequences: A224243 A224244 A224245 * A224247 A224248 A224249

Keyword

nonn

Author

Geoffrey Critzer, Apr 01 2013

Status

approved