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A362679 a(n) is the permanent of the n X n symmetric matrix M(n) defined by M[i, j, n] = min(i, j)*(n + 1) - i*j.

%I A362679 #37 May 25 2023 07:12:04
%S A362679 1,1,5,72,2309,140400,14495641,2347782144,562385930985,
%T A362679 190398813728000,87889475202276461,53726132414026874880,
%U A362679 42454821207656237294381,42495322215073539046387712,52954624815227996007075890625,80932107560443542398970529579008,149736953621087625813286348913927569
%N A362679 a(n) is the permanent of the n X n symmetric matrix M(n) defined by M[i, j, n] = min(i, j)*(n + 1) - i*j.
%C A362679 M(n-1)/n is the inverse of the Cartan matrix for SU(n): the special unitary group of degree n.
%C A362679 The elements sum of the matrix M(n) is A002415(n+1).
%C A362679 The antidiagonal sum of the matrix M(n) is A005993(n-1).
%C A362679 The n-th row of A107985 gives the row or column sums of the matrix M(n+1).
%D A362679 E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Am. Math. Soc. Translations, Series 2, Vol. 6, 1957.
%H A362679 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 A362679 Stefano Spezia, <a href="/A362679/a362679.pdf">A determinantal formula for the number of trees on n labeled nodes</a>
%H A362679 Wikipedia, <a href="https://en.wikipedia.org/wiki/Special_unitary_group">Special unitary group</a>
%F A362679 Conjecture: det(M(n)) = A000272(n+1).
%F A362679 The conjecture is true (see proof in Links). - _Stefano Spezia_, May 24 2023
%e A362679 a(3) = 72:
%e A362679            [3, 2, 1]
%e A362679     M(3) = [2, 4, 2]
%e A362679            [1, 2, 3]
%e A362679 a(5) = 140400:
%e A362679            [5, 4, 3, 2, 1]
%e A362679            [4, 8, 6, 4, 2]
%e A362679     M(5) = [3, 6, 9, 6, 3]
%e A362679            [2, 4, 6, 8, 4]
%e A362679            [1, 2, 3, 4, 5]
%p A362679 a:= n-> `if`(n=0, 1, LinearAlgebra[Permanent](
%p A362679     Matrix(n, (i, j)-> min(i, j)*(n+1)-i*j))):
%p A362679 seq(a(n), n=0..16);  # _Alois P. Heinz_, Apr 30 2023
%t A362679 M[i_, j_, n_]:=Min[i, j](n+1)-i j; Join[{1}, Table[Permanent[Table[M[i, j, n], {i, n}, {j, n}]], {n, 17}]]
%o A362679 (PARI) a(n) = matpermanent(matrix(n, n, i, j, min(i, j)*(n + 1) - i*j)); \\ _Michel Marcus_, Apr 30 2023
%Y A362679 Cf. A000272, A000292 (trace), A002415, A003983, A003991, A005993, A106314 (antidiagonals), A107985.
%K A362679 nonn
%O A362679 0,3
%A A362679 _Stefano Spezia_, Apr 29 2023