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%I #14 Sep 28 2022 08:13:10
%S 1,1,8,216,7344,168183,7226091,295259094,11801772252,1673511251940,
%T 65568867621336,2710049208604776,202103867012027328,
%U 12881755844526953376,736186737257150962752,70484099228399057425344,5507570249593121504026368,434305172863416192470350848,122043063804581668929348667392
%N a(n) is the permanent of the n X n symmetric matrix M(n) whose generic element M[i,j] is equal to the digital root of i*j.
%C The matrix M(n) is nonsingular only for n = 1, 5 and 6 with determinant equal respectively to 1, 6561 and 59049.
%C The rank of M(n) is 1 for 1 <= n <= 3, 3 for n = 4, 5 for n = 5, 6 for 6 <= n <= 8, and 7 for n >= 9. - _Jianing Song_, Sep 28 2022
%F Sum_{i=1..n} M[n-i+1,i] = A353128(n+1).
%e a(7) = 7226091:
%e 1, 2, 3, 4, 5, 6, 7
%e 2, 4, 6, 8, 1, 3, 5
%e 3, 6, 9, 3, 6, 9, 3
%e 4, 8, 3, 7, 2, 6, 1
%e 5, 1, 6, 2, 7, 3, 8
%e 6, 3, 9, 6, 3, 9, 6
%e 7, 5, 3, 1, 8, 6, 4
%t M[i_, j_]:=If[i*j==0, 0, 1+Mod[i*j-1, 9]]; Join[{1},Table[Permanent[Table[M[i, j], {i, n}, {j, n}]],{n,18}]]
%o (PARI) a(n) = matpermanent(matrix(n, n, i, j, (i*j-1)%9+1)); \\ _Michel Marcus_, May 12 2022
%Y Cf. A003991, A010888, A353109, A353128, A353974 (trace of the matrix M(n)).
%K nonn,base
%O 0,3
%A _Stefano Spezia_, May 11 2022
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