A357421 a(n) is the hafnian of the 2n X 2n symmetric matrix whose generic element M[i,j] is equal to the digital root of i*j.

1, 2, 54, 1377, 55350, 4164534, 217595322, 11974135554, 999599777190, 150051627647010, 11873389098337236

Offset

0,2

Links

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

Wikipedia, Hafnian

Wikipedia, Symmetric matrix

Example

a(3) = 1377:
    1, 2, 3, 4, 5, 6;
    2, 4, 6, 8, 1, 3;
    3, 6, 9, 3, 6, 9;
    4, 8, 3, 7, 2, 6;
    5, 1, 6, 2, 7, 3;
    6, 3, 9, 6, 3, 9.

Mathematica

M[i_, j_, n_] := If[i*j == 0, 0, 1 + Mod[i*j - 1, 9]]; 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. A003991, A010888, A353109, A353933 (permanent of M(n)), A353974 (trace of M(n)).

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

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

Sequence in context: A057411 A157058 A305693 * A071798 A338514 A123686

Adjacent sequences: A357418 A357419 A357420 * A357422 A357423 A357424

Keyword

nonn,base,hard,more

Author

Stefano Spezia, Sep 27 2022

Extensions

a(6)-a(10) from Pontus von Brömssen, Oct 15 2023

Status

approved