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A338456 a(n) is the hafnian of a symmetric Toeplitz matrix M(2n) whose first row consists of a single zero followed by successive positive integers repeated (A004526). 11
1, 1, 4, 45, 968, 34265, 1799748, 131572357, 12770710096, 1589142683313, 246658484353100 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Wikipedia, Hafnian
Wikipedia, Symmetric matrix
Wikipedia, Toeplitz Matrix
EXAMPLE
a(2) = 4 because the hafnian of
0 1 1 2
1 0 1 1
1 1 0 1
2 1 1 0
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 4.
MATHEMATICA
k[i_]:=Floor[i/2]; M[i_, j_, n_]:=Part[Part[ToeplitzMatrix[Array[k, 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, 5, 0]
PROG
(PARI) tm(n) = {my(m = matrix(n, n, i, j, if (i==1, j\2, if (j==1, i\2)))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m; }
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, Nov 11 2020
CROSSREFS
Cf. A004526.
Cf. A002378 (conjectured determinant of M(2n+1)), A083392 (conjectured determinant of M(n+1)), A332566 (permanent of M(n)), A333119 (k-th super- and subdiagonal sums of the matrix M(n)).
Sequence in context: A214400 A360344 A197989 * A276292 A174484 A158887
KEYWORD
nonn,hard,more
AUTHOR
Stefano Spezia, Oct 28 2020
EXTENSIONS
a(5) from Michel Marcus, Nov 11 2020
a(6)-a(10) from Pontus von Brömssen, Oct 14 2023
STATUS
approved

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Last modified May 24 15:36 EDT 2024. Contains 372778 sequences. (Running on oeis4.)