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%I #28 Oct 15 2023 09:27:03
%S 1,1,17,1177,210249,76961257,50203153993,53127675356625,
%T 85252003916011889,197131843368693693937,631233222450168374457057
%N a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i, j] = min(i, j)*(2*n + 1) - i*j.
%C M(n-1)/n is the inverse of the Cartan matrix for SU(n): the special unitary group of degree n.
%C The elements sum of the matrix M(n) is A002415(n+1).
%C The antidiagonal sum of the matrix M(n) is A005993(n-1).
%C The n-th row of A107985 gives the row or column sums of the matrix M(n+1).
%D E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Am. Math. Soc. Translations, Series 2, Vol. 6, 1957.
%H Chao Ju, <a href="https://arxiv.org/abs/2304.11830">Chern-Simons Theory, Ehrhart Polynomials, and Representation Theory</a>, arXiv:2304.11830 [math-ph], 2023. See p. 14.
%H Stefano Spezia, <a href="/A362679/a362679.pdf">A determinantal formula for the number of trees on n labeled nodes</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hafnian">Hafnian</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Special_unitary_group">Special unitary group</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_matrix">Symmetric matrix</a>
%F Conjecture: det(M(n)) = A000272(n+1).
%F The conjecture is true (see proof in Links). - _Stefano Spezia_, May 24 2023
%e a(2) = 17:
%e [4, 3, 2, 1]
%e [3, 6, 4, 2]
%e [2, 4, 6, 3]
%e [1, 2, 3, 4]
%t M[i_, j_, n_]:=Part[Part[Table[Min[r,c](n+1)-r c, {r, n}, {c, n}], i], j]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i], 2n], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 6, 0]
%o (PARI) tm(n) = matrix(n, n, i, j, min(i, j)*(n + 1) - i*j);
%o a(n) = my(m = tm(2*n), s=0); forperm([1..2*n], p, s += prod(j=1, n, m[p[2*j-1], p[2*j]]); ); s/(n!*2^n); \\ _Michel Marcus_, May 02 2023
%Y Cf. A000272, A000292 (trace), A002415, A003983, A003991, A005993, A106314 (antidiagonals), A107985, A362679 (permanent).
%K nonn,hard,more
%O 0,3
%A _Stefano Spezia_, Apr 30 2023
%E a(6) from _Michel Marcus_, May 02 2023
%E a(7)-a(10) from _Pontus von Brömssen_, Oct 15 2023
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