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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.
3
1, 1, 5, 72, 2309, 140400, 14495641, 2347782144, 562385930985, 190398813728000, 87889475202276461, 53726132414026874880, 42454821207656237294381, 42495322215073539046387712, 52954624815227996007075890625, 80932107560443542398970529579008, 149736953621087625813286348913927569
OFFSET
0,3
COMMENTS
M(n-1)/n is the inverse of the Cartan matrix for SU(n): the special unitary group of degree n.
The elements sum of the matrix M(n) is A002415(n+1).
The antidiagonal sum of the matrix M(n) is A005993(n-1).
The n-th row of A107985 gives the row or column sums of the matrix M(n+1).
REFERENCES
E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Am. Math. Soc. Translations, Series 2, Vol. 6, 1957.
FORMULA
Conjecture: det(M(n)) = A000272(n+1).
The conjecture is true (see proof in Links). - Stefano Spezia, May 24 2023
EXAMPLE
a(3) = 72:
[3, 2, 1]
M(3) = [2, 4, 2]
[1, 2, 3]
a(5) = 140400:
[5, 4, 3, 2, 1]
[4, 8, 6, 4, 2]
M(5) = [3, 6, 9, 6, 3]
[2, 4, 6, 8, 4]
[1, 2, 3, 4, 5]
MAPLE
a:= n-> `if`(n=0, 1, LinearAlgebra[Permanent](
Matrix(n, (i, j)-> min(i, j)*(n+1)-i*j))):
seq(a(n), n=0..16); # Alois P. Heinz, Apr 30 2023
MATHEMATICA
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}]]
PROG
(PARI) a(n) = matpermanent(matrix(n, n, i, j, min(i, j)*(n + 1) - i*j)); \\ Michel Marcus, Apr 30 2023
CROSSREFS
Cf. A000272, A000292 (trace), A002415, A003983, A003991, A005993, A106314 (antidiagonals), A107985.
Sequence in context: A282788 A197324 A197977 * A307932 A136637 A319767
KEYWORD
nonn
AUTHOR
Stefano Spezia, Apr 29 2023
STATUS
approved