A357420 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = abs(i - j) if min(i, j) < max(i, j) <= 2*min(i, j), and otherwise 0.

1, 1, 1, 8, 86, 878, 13730, 348760, 11622396, 509566864, 26894616012, 1701189027944, 125492778658096, 10738546182981256, 1049631636279244832, 117756049412699967072

Offset

0,4

Links

Table of n, a(n) for n=0..15.

Wikipedia, Hafnian

Wikipedia, Symmetric matrix

Example

a(4) = 86:
    0,  1,  0,  0,  0,  0,  0,  0;
    1,  0,  1,  2,  0,  0,  0,  0;
    0,  1,  0,  1,  2,  3,  0,  0;
    0,  2,  1,  0,  1,  2,  3,  4;
    0,  0,  2,  1,  0,  1,  2,  3;
    0,  0,  3,  2,  1,  0,  1,  2;
    0,  0,  0,  3,  2,  1,  0,  1;
    0,  0,  0,  4,  3,  2,  1,  0.

Mathematica

M[i_, j_, n_] := If[Min[i, j] < Max[i, j] <= 2 Min[i, j], Abs[j - i], 0]; a[n_] := Sum[Product[M[Part[PermutationList[s, 2 n], 2 i - 1], Part[PermutationList[s, 2 n], 2 i], 2 n], {i, n}], {s, SymmetricGroup[2 n] // GroupElements}]/(n!*2^n); Array[a, 6, 0]

Crossrefs

Cf. A000982 (number of zero matrix elements of M(n)), A003983, A007590 (number of positive matrix elements of M(n)), A049581, A051125, A352967, A353452 (determinant of M(n)), A353453 (permanent of M(n)).

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

Cf. A356481, A356482, A356483, A356484, A357279.

Sequence in context: A371897 A180582 A230621 * A371407 A369505 A268052

Adjacent sequences: A357417 A357418 A357419 * A357421 A357422 A357423

Keyword

nonn,hard,more

Author

Stefano Spezia, Sep 27 2022

Extensions

a(6)-a(15) from Pontus von Brömssen, Oct 16 2023

Status

approved