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A274868 Number of set partitions of [n] into exactly four blocks such that all odd elements are in blocks with an odd index and all even elements are in blocks with an even index.

%I A274868 #16 Jul 15 2023 20:52:58
%S A274868 1,2,7,14,35,70,155,310,651,1302,2667,5334,10795,21590,43435,86870,
%T A274868 174251,348502,698027,1396054,2794155,5588310,11180715,22361430,
%U A274868 44731051,89462102,178940587,357881174,715795115,1431590230,2863245995,5726491990,11453115051
%N A274868 Number of set partitions of [n] into exactly four blocks such that all odd elements are in blocks with an odd index and all even elements are in blocks with an even index.
%H A274868 Alois P. Heinz, <a href="/A274868/b274868.txt">Table of n, a(n) for n = 4..1000</a>
%H A274868 Pedro Fernando Fernández Espinosa and Agustín Moreno Cañadas, <a href="https://arxiv.org/abs/2104.00050">Homological Ideals as Integer Specializations of Some Brauer Configuration Algebras</a>, arXiv:2104.00050 [math.RT], 2021.
%F A274868 G.f.: -x^4/((x-1)*(2*x-1)*(x+1)*(2*x^2-1)).
%F A274868 From _Ridouane Oudra_, Jul 13 2023: (Start)
%F A274868 a(n) = x/6 + ((-1)^n - 1)*y, where x = 2^n - 3*sqrt(2)^n + 2 and y = (1/2)*sqrt(2)^(n-1) - (1/4)*sqrt(2)^n - 1/6.
%F A274868 a(n) = ((3 - (-1)^n)/12)*(4^floor(n/2) - 3*2^floor(n/2) + 2).
%F A274868 a(2n) = (4^n)/6 - 2^(n-1) + 1/3.
%F A274868 a(2n+1) = 2*a(2n). (End)
%e A274868 a(6) = 7: 13|24|5|6, 15|24|3|6, 1|24|35|6, 15|26|3|4, 15|2|3|46, 1|26|35|4, 1|2|35|46.
%e A274868 a(7) = 14: 137|24|5|6, 13|24|57|6, 157|24|3|6, 15|24|37|6, 17|24|35|6, 1|24|357|6, 157|26|3|4, 15|26|37|4, 157|2|3|46, 15|2|37|46, 17|26|35|4, 1|26|357|4, 17|2|35|46, 1|2|357|46.
%t A274868 Drop[CoefficientList[Series[-x^4/((x - 1) (2 x - 1) (x + 1) (2 x^2 - 1)), {x, 0, 36}], x], 4] (* _Michael De Vlieger_, Jun 15 2021 *)
%Y A274868 Column k=4 of A274537.
%K A274868 nonn,easy
%O A274868 4,2
%A A274868 _Alois P. Heinz_, Jul 09 2016