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A351869 a(n) is the number of self-complementary score sequences that are possible for strong tournaments on n vertices.

%I A351869 #36 Dec 25 2024 04:06:06
%S A351869 1,1,0,1,1,3,3,9,11,30,39,103,141,363,514,1301,1894,4727,7036,17358,
%T A351869 26311,64282,98936,239712,373806,899115,1418130,3389078,5399133,
%U A351869 12828749,20619565,48739465,78963217,185769110,303128971,710067027,1166206802,2720959217,4495461790
%N A351869 a(n) is the number of self-complementary score sequences that are possible for strong tournaments on n vertices.
%C A351869 See A000571 for the definition of a tournament and its score sequence. A tournament is strong if every two vertices are mutually reachable by directed paths. Alternatively, a tournament is strong if it contains a directed Hamiltonian cycle.
%C A351869 A sequence (s_1 <= s_2 <= ... <= s_n) of integers is the score sequence of a strong tournament iff Sum_{i=1..r} s_i > binomial(r,2) for 1 <= r < n and Sum_{i=1..n} s_i = binomial(n,2). It is self-complementary iff s_{n+1-i} = n-1-s_i for 1 <= i <= n/2.
%H A351869 Paul K. Stockmeyer, <a href="/A351869/b351869.txt">Table of n, a(n) for n = 0..501</a>
%H A351869 Paul K. Stockmeyer, <a href="http://arxiv.org/abs/2202.05238">Counting Various Classes of Tournament Score Sequences</a>, arXiv:2202.05238 [math.CO], 2022.
%H A351869 Paul K. Stockmeyer, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Stockmeyer/stock14.html">Counting Various Classes of Tournament Score Sequences</a>, J. Integer Seq. 26 (2023), Article 23.5.2.
%F A351869 For n >= 1, a(2*n) = Sum_{T=binomial(n,2)+1..n*(n-1)} Sum_{E=floor(n/2)..n-1} g_n(T,E) and a(2*n+1) = Sum_{T=binomial(n,2)+1..n^2} Sum_{E=floor(n/2)..n} g_n(T,E), where g_1(T, E)=[T=E], and for n>=2, g_n(T, E)=0 if T-E <= binomial(n-1, 2) and g_n(T, E) = Sum_{k=0..E} g_{n-1}(T-E, k) otherwise.
%e A351869 The 9 self-complementary strong score sequences of length seven are
%e A351869   (1,1,2,3,4,5,5),
%e A351869   (1,1,3,3,3,5,5),
%e A351869   (1,2,2,3,4,4,5),
%e A351869   (1,2,3,3,3,4,5),
%e A351869   (1,3,3,3,3,3,5),
%e A351869   (2,2,2,3,4,4,4),
%e A351869   (2,2,3,3,3,4,4),
%e A351869   (2,3,3,3,3,3,4),
%e A351869   (3,3,3,3,3,3,3).
%Y A351869 Cf. A000571, A351822, A345470.
%K A351869 nonn
%O A351869 0,6
%A A351869 _Paul K. Stockmeyer_, Feb 22 2022