A362711 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.

1, 1, 17, 1177, 210249, 76961257, 50203153993, 53127675356625, 85252003916011889, 197131843368693693937, 631233222450168374457057

Offset: 0

Author

Stefano Spezia, Apr 30 2023

Keywords

• nonn
• hard
• more

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).

Examples

			a(2) = 17:
    [4, 3, 2, 1]
    [3, 6, 4, 2]
    [2, 4, 6, 3]
    [1, 2, 3, 4]
		

References

• E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Am. Math. Soc. Translations, Series 2, Vol. 6, 1957.

Links

• Chao Ju, Chern-Simons Theory, Ehrhart Polynomials, and Representation Theory, arXiv:2304.11830 [math-ph], 2023. See p. 14.
• Stefano Spezia, A determinantal formula for the number of trees on n labeled nodes
• Wikipedia, Hafnian
• Wikipedia, Special unitary group
• Wikipedia, Symmetric matrix

Crossrefs

Cf. A000272, A000292 (trace), A002415, A003983, A003991, A005993, A106314 (antidiagonals), A107985, A362679 (permanent).

Programs

Mathematica

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]

PARI

tm(n) = matrix(n, n, i, j, min(i, j)*(n + 1) - i*j);
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

Formula

Conjecture: det(M(n)) = A000272(n+1).

The conjecture is true (see proof in Links). - Stefano Spezia, May 24 2023

Extensions

a(6) from Michel Marcus, May 02 2023

a(7)-a(10) from Pontus von Brömssen, Oct 15 2023