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A204249 Permanent of the n-th principal submatrix of A003057.

%I A204249 #40 Jun 27 2025 22:12:56
%S A204249 1,2,17,336,12052,685080,56658660,6428352000,958532774976,
%T A204249 181800011433600,42745508545320000,12203347213269273600,
%U A204249 4158410247782904833280,1667267950805177583582720,776990110000329481864608000,416483579190482716042690560000
%N A204249 Permanent of the n-th principal submatrix of A003057.
%C A204249 I have proved that for any odd prime p we have a(p) == p (mod p^2). - _Zhi-Wei Sun_, Aug 30 2021
%H A204249 Vaclav Kotesovec, <a href="/A204249/b204249.txt">Table of n, a(n) for n = 0..36</a>
%H A204249 Zhi-Wei Sun, <a href="https://arxiv.org/abs/2108.07723">Arithmetic properties of some permanents</a>, arXiv:2108.07723 [math.GM], 2021.
%F A204249 From _Vaclav Kotesovec_, Dec 01 2016: (Start)
%F A204249 a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = A278300 = 2.455407482284127949... and c = 1.41510164826...
%F A204249 a(n) ~ c * d^n * n^(2*n + 1/2), where d = A278300/exp(2) = 0.332303267076220516... and c = 8.89134588451...
%F A204249 (End)
%p A204249 with(LinearAlgebra):
%p A204249 a:= n-> `if`(n=0, 1, Permanent(Matrix(n, (i, j)-> i+j))):
%p A204249 seq(a(n), n=0..16);  # _Alois P. Heinz_, Nov 14 2016
%t A204249 f[i_, j_] := i + j;
%t A204249 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204249 TableForm[m[8]] (* 8x8 principal submatrix *)
%t A204249 Flatten[Table[f[i, n + 1 - i],
%t A204249   {n, 1, 12}, {i, 1, n}]]  (* A003057 *)
%t A204249 Permanent[m_] :=
%t A204249   With[{a = Array[x, Length[m]]},
%t A204249    Coefficient[Times @@ (m.a), Times @@ a]];
%t A204249 Table[Permanent[m[n]], {n, 1, 15}]  (* A204249 *)
%o A204249 (PARI) {a(n) = matpermanent(matrix(n, n, i, j, i+j))}
%o A204249 for(n=0, 20, print1(a(n), ", ")) \\ _Vaclav Kotesovec_, Dec 21 2018
%Y A204249 Cf. A000442, A003057, A085750, A085807, A278845, A278847, A278300, A278925, A278926.
%K A204249 nonn
%O A204249 0,2
%A A204249 _Clark Kimberling_, Jan 14 2012
%E A204249 a(0)=1 prepended and one more term added by _Alois P. Heinz_, Nov 14 2016