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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.

%I A357421 #11 Oct 15 2023 09:26:39
%S A357421 1,2,54,1377,55350,4164534,217595322,11974135554,999599777190,
%T A357421 150051627647010,11873389098337236
%N 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.
%H A357421 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hafnian">Hafnian</a>
%H A357421 Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_matrix">Symmetric matrix</a>
%e A357421 a(3) = 1377:
%e A357421     1, 2, 3, 4, 5, 6;
%e A357421     2, 4, 6, 8, 1, 3;
%e A357421     3, 6, 9, 3, 6, 9;
%e A357421     4, 8, 3, 7, 2, 6;
%e A357421     5, 1, 6, 2, 7, 3;
%e A357421     6, 3, 9, 6, 3, 9.
%t A357421 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]
%Y A357421 Cf. A003991, A010888, A353109, A353933 (permanent of M(n)), A353974 (trace of M(n)).
%Y A357421 Cf. A202038, A336114, A336286, A336400, A338456.
%Y A357421 Cf. A356481, A356482, A356483, A356484, A357279.
%K A357421 nonn,base,hard,more
%O A357421 0,2
%A A357421 _Stefano Spezia_, Sep 27 2022
%E A357421 a(6)-a(10) from _Pontus von Brömssen_, Oct 15 2023