%I #39 Mar 05 2022 00:23:24
%S 2,10,29,66,129,228,374,580,860,1230,1707,2310,3059,3976,5084,6408,
%T 7974,9810,11945,14410,17237,20460,24114,28236,32864,38038,43799,
%U 50190,57255,65040,73592,82960,93194,104346,116469,129618,143849,159220,175790,193620
%N a(n) is the number of 2-point antichains in the poset D_{2n+1} of type D, whose elements are compositions of 2n+1.
%C A type D poset has elements which are compositions with four parts {x_{1}, x_{2}, x_{3}, x_{4}} satisfying:
%C x_{i} >= 0 for i = 1..4,
%C at least two of its elements are positive,
%C x_{2} = x_{4}, and x_{3} - x_{1} >= 0.
%C The partial order relation '<' is defined as follows: {x_{1}, x_{2}, x_{3}, x_{4}} < {x'_{1}, x'_{2}, x'_{3}, x'_{4}} if and only if x'_{1} <= x_{1}, x_{2} <= x'_{2}, x'_{3} <= x_{3}, and x_{4} <= x'_{4}.
%C The poset D_{2n+1} is a poset of type D where the compositions belonging to D_{2n+1} are compositions of the number 2n+1, with n >= 2. A nice property of this type of poset is that the Hasse diagram of the poset D_{2n+1} is equal to the Hasse diagram of the poset D_{2n}.
%H Michael De Vlieger, <a href="/A344791/b344791.txt">Table of n, a(n) for n = 2..10000</a>
%H Natalia Agudelo Muñetón, Agustín Moreno Cañadas, Pedro Fernando Fernández Espinosa, and Isaías David Marín Gaviria, <a href="https://doi.org/10.3390/math9233042">Brauer Configuration Algebras and Their Applications in Graph Energy Theory</a>, Mathematics (2021) Vol. 9, 3042.
%H I. D. M. Gaviria, <a href="https://repositorio.unal.edu.co/handle/unal/78154">The Auslander-Reiten Quiver of Equipped Posets of Finite Growth Representation Type, some Functorial Descriptions and Its Applications</a>, (Dissertation) Universidad Nacional de Colombia, 2020, 164 pp.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,0,5,-4,1).
%F a(n) = Sum_{i=1..n} Sum_{j=0..floor(i/2)} h(n,i,j)*(t(i)-2*t(j)), where h(n,i,j) = 0 if i=n and j=0; h(n,i,j) = n+1-i if i=2j and 1 <= j <= floor(n/2); h(n,i,j) = 1 otherwise. n >= 2 and t(i) = A000217(i) is the i-th triangular number.
%F G.f.: x^2*(x^2-2*x-2)/((x+1)*(x-1)^5). - _Alois P. Heinz_, May 29 2021
%F a(n) = (1 - (-1)^n - 8*n - 4*n^2 + 8*n^3 + 2*n^4)/32. - _Stefano Spezia_, Jun 02 2021
%e For instance, the first term of the sequence is for n=2: a(2)=2 means that there are two 2-point antichains in the poset D_{5}={{1,0,4,0},{2,0,3,0},{0,1,3,1},{0,2,1,2},{1,1,2,1}}, namely ({1,0,4,0}, {2,0,3,0}) and ({0,1,3,1}, {1,1,2,1}).
%t Drop[CoefficientList[Series[x^2*(x^2 - 2 x - 2)/((x + 1) (x - 1)^5), {x, 0, 41}], x], 2] (* _Michael De Vlieger_, Mar 04 2022 *)
%o (PARI) a(n) = (n-1)*(n+1)*(n^2+4*n-1) >> 4; \\ _Kevin Ryde_, May 29 2021
%Y Cf. A000217.
%K nonn,easy
%O 2,1
%A _Isaías David Marín Gaviria_ and Agustín Moreno Cañadas, May 28 2021