A356482 a(n) is the hafnian of a symmetric Toeplitz matrix M(2*n) whose first row consists of 2*n, 2*n-1, ..., 1.

1, 1, 16, 714, 62528, 9056720, 1960138560, 592615689904, 238560786221056, 123358665203311104, 79683847063011614720

Offset

0,3

Links

Table of n, a(n) for n=0..10.

Wikipedia, Hafnian

Wikipedia, Symmetric matrix

Wikipedia, Toeplitz Matrix

Example

a(2) = 16 because the hafnian of
    4  3  2  1
    3  4  3  2
    2  3  4  3
    1  2  3  4
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 16.

Mathematica

k[i_]:=i; M[i_, j_, n_]:=Part[Part[ToeplitzMatrix[Reverse[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, 6, 0]

Prog

(PARI) tm(n) = my(m = matrix(n, n, i, j, if (i==1, n-j+1, if (j==1, n-i+1)))); 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, May 02 2023

Crossrefs

Cf. A001792 (determinant of M(n)), A307783.

Cf. A202038, A336114, A336286, A336400, A338456.

Cf. A356481, A356483, A356484.

Sequence in context: A283534 A294704 A264114 * A201622 A362574 A220809

Adjacent sequences: A356479 A356480 A356481 * A356483 A356484 A356485

Keyword

nonn,hard,more

Author

Stefano Spezia, Aug 09 2022

Extensions

a(6) from Michel Marcus, May 02 2023

a(7)-a(10) from Pontus von Brömssen, Oct 14 2023

Status

approved