A032262 Number of ways to partition n labeled elements into pie slices allowing the pie to be turned over.

1, 1, 2, 5, 17, 83, 557, 4715, 47357, 545963, 7087517, 102248075, 1622633597, 28091569643, 526858352477, 10641342978635, 230283190994237, 5315654682014123, 130370767029201437, 3385534663256976395

Offset

0,3

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200

C. G. Bower, Transforms (2)

Formula

a(n) = 2^(n-2) + A000670(n-1) for n >= 2. - N. J. A. Sloane, Jan 17 2008

a(n) = 2^(n-1) + Sum_{k >= 3} Stirling_2(n,k)*(k-1)!/2 for n >= 1. - N. J. A. Sloane, Jan 17 2008

"DIJ" (bracelet, indistinct, labeled) transform of 1, 1, 1, 1, ... (see Bower link).

E.g.f.: 1 + (g(x) + g(x)^2/2 - log(1-g(x)))/2 where g(x) = exp(x) - 1. - Andrew Howroyd, Sep 12 2018

Example

For n = 4 we have the following "pies":
. 1
./ \
2 . 3 . 12 .. 12 . 123 .1234
.\ / .. / \ .(..)..(..)
. 4 .. 3--4 . 34 .. 4
.(3)....(6)...(3)..(4)...(1) Total a(4) = 17

Mathematica

a[0] = a[1] = 1; a[n_] := 2^(n-2) + HurwitzLerchPhi[1/2, 1-n, 0]/2;
Array[a, 20, 0] (* Jean-François Alcover, Aug 26 2019 *)

Prog

(PARI) seq(n)={my(p=exp(x + O(x*x^n))-1); Vec(1 + serlaplace(p + p^2/2 - log(1-p))/2)} \\ Andrew Howroyd, Sep 12 2018

Crossrefs

Row sums of triangle A133800.

Sequence in context: A098540 A079574 A363002 * A144259 A191799 A079805

Adjacent sequences: A032259 A032260 A032261 * A032263 A032264 A032265

Keyword

nonn

Author

Christian G. Bower

Extensions

Edited by N. J. A. Sloane, Jan 17 2008

a(0)=1 prepended by Andrew Howroyd, Sep 12 2018

Status

approved