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

%I A032262 #21 Aug 26 2019 11:46:30
%S A032262 1,1,2,5,17,83,557,4715,47357,545963,7087517,102248075,1622633597,
%T A032262 28091569643,526858352477,10641342978635,230283190994237,
%U A032262 5315654682014123,130370767029201437,3385534663256976395
%N A032262 Number of ways to partition n labeled elements into pie slices allowing the pie to be turned over.
%H A032262 Andrew Howroyd, <a href="/A032262/b032262.txt">Table of n, a(n) for n = 0..200</a>
%H A032262 C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%F A032262 a(n) = 2^(n-2) + A000670(n-1) for n >= 2. - _N. J. A. Sloane_, Jan 17 2008
%F A032262 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
%F A032262 "DIJ" (bracelet, indistinct, labeled) transform of 1, 1, 1, 1, ... (see Bower link).
%F A032262 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
%e A032262 For n = 4 we have the following "pies":
%e A032262 . 1
%e A032262 ./ \
%e A032262 2 . 3 . 12 .. 12 . 123 .1234
%e A032262 .\ / .. / \ .(..)..(..)
%e A032262 . 4 .. 3--4 . 34 .. 4
%e A032262 .(3)....(6)...(3)..(4)...(1) Total a(4) = 17
%t A032262 a[0] = a[1] = 1; a[n_] := 2^(n-2) + HurwitzLerchPhi[1/2, 1-n, 0]/2;
%t A032262 Array[a, 20, 0] (* _Jean-François Alcover_, Aug 26 2019 *)
%o A032262 (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
%Y A032262 Row sums of triangle A133800.
%K A032262 nonn
%O A032262 0,3
%A A032262 _Christian G. Bower_
%E A032262 Edited by _N. J. A. Sloane_, Jan 17 2008
%E A032262 a(0)=1 prepended by _Andrew Howroyd_, Sep 12 2018