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

Author

Christian G. Bower

Keywords

• nonn

Examples

			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
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200
• C. G. Bower, Transforms (2)

Crossrefs

Row sums of triangle A133800.

Programs

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 *)

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

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

Extensions

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

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