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A204235 Permanent of the n-th principal submatrix of A143182.

%I A204235 #28 Apr 30 2019 02:31:54
%S A204235 1,5,42,632,14124,449652,19200336,1063272704,74068997888,
%T A204235 6344884818304,655635015988864,80447681129070080,11565193558509497088,
%U A204235 1925787312858332101888,367762470538537620457472,79847718328265949957881856,19560087897336150724249288704
%N A204235 Permanent of the n-th principal submatrix of A143182.
%C A204235 The n-th principal submatrix of A143182 is an n X n symmetric Toeplitz matrix whose first row consists of successive natural numbers 1, ..., n. - _Stefano Spezia_, Sep 23 2018
%C A204235 Conjecture: a(1) and a(2) are the only terms that are odd numbers. - _Stefano Spezia_, Oct 28 2018
%H A204235 Vaclav Kotesovec, <a href="/A204235/b204235.txt">Table of n, a(n) for n = 1..35</a> (terms 1..14 from Clark Kimberling, terms 15..22 from Stefano Spezia)
%p A204235 f:= proc(n) uses LinearAlgebra;
%p A204235 Permanent(ToeplitzMatrix([seq(i, i=1 ..n)], n, symmetric))
%p A204235 end proc:
%p A204235 map(f, [$1..20]); # _Stefano Spezia_, Oct 28 2018
%t A204235 f[i_, j_] := Max[i - j + 1, j - i + 1];  (* A143182 *)
%t A204235 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204235 TableForm[m[4]] (* 4 X 4 principal submatrix *)
%t A204235 Table[Det[m[n]], {n, 1, 22}]  (* A001792 - signed *)
%t A204235 Permanent[m_] :=
%t A204235   With[{a = Array[x, Length[m]]},
%t A204235    Coefficient[Times @@ (m.a), Times @@ a]];
%t A204235 Table[Permanent[m[n]], {n, 1, 14}]  (* A204235 *)
%t A204235 b[i_]:=i; a[n_]:=Permanent[ToeplitzMatrix[Array[b, n], Array[b, n]]]; Array[a, 22] (* _Stefano Spezia_, Sep 23 2018 *)
%o A204235 (PARI) {a(n) = matpermanent(matrix(n, n, i, j, max(i - j + 1, j - i + 1)))}
%o A204235 for(n=1, 20, print1(a(n), ", ")) \\ _Vaclav Kotesovec_, Apr 29 2019
%Y A204235 Cf. A143182.
%K A204235 nonn
%O A204235 1,2
%A A204235 _Clark Kimberling_, Jan 13 2012
%E A204235 Extended by _Stefano Spezia_, Oct 28 2018