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A032278 Number of ways to partition n elements into pie slices each with at least 2 elements allowing the pie to be turned over.

%I A032278 #22 Feb 21 2025 05:47:04
%S A032278 0,1,1,2,2,4,4,7,8,13,15,25,30,48,63,98,132,208,290,454,656,1021,1509,
%T A032278 2358,3544,5535,8441,13200,20318,31835,49352,77435,120710,189673,
%U A032278 296853,467159,733362,1155646,1818593,2869377,4524080
%N A032278 Number of ways to partition n elements into pie slices each with at least 2 elements allowing the pie to be turned over.
%H A032278 Andrew Howroyd, <a href="/A032278/b032278.txt">Table of n, a(n) for n = 1..500</a>
%H A032278 C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%F A032278 "DIK" (bracelet, indistinct, unlabeled) transform of 0, 1, 1, 1, ...
%F A032278 G.f.: (x^2/((1 - x)*(1 - x^2 - x^4)) + Sum_{d>0} phi(d)*log((1 - x^d)/(1 - x^d - x^(2*d)))/d)/2. - _Andrew Howroyd_, Jun 20 2018
%p A032278 A032278_list := proc(n)
%p A032278     local ele ;
%p A032278     ele := [0,seq(1,i=1..30)] ;
%p A032278     DIK(ele) ; # defined in A032287
%p A032278 end proc:
%p A032278 A032278_list(50) ; # _R. J. Mathar_, Feb 14 2025
%o A032278 (PARI) seq(n)={Vec(x^2/((1-x)*(1-x^2-x^4)) + sum(d=1, n, eulerphi(d)/d*log((1-x^d)/(1-x^d-x^(2*d)) + O(x*x^n))), -n)/2} \\ _Andrew Howroyd_, Jun 20 2018
%Y A032278 Cf. A000005, A027750, A032190 (CIK, pie not to be turned over).
%K A032278 nonn
%O A032278 1,4
%A A032278 _Christian G. Bower_