%I #38 Aug 30 2021 21:36:13

%S 1,2,17,336,12052,685080,56658660,6428352000,958532774976,

%T 181800011433600,42745508545320000,12203347213269273600,

%U 4158410247782904833280,1667267950805177583582720,776990110000329481864608000,416483579190482716042690560000

%N Permanent of the n-th principal submatrix of A003057.

%C I have proved that for any odd prime p we have a(p) == p (mod p^2). - _Zhi-Wei Sun_, Aug 30, 2021

%H Vaclav Kotesovec, <a href="/A204249/b204249.txt">Table of n, a(n) for n = 0..36</a>

%H Zhi-Wei Sun, <a href="https://arxiv.org/abs/2108.07723">Arithmetic properties of some permanents</a>, arXiv:2108.07723 [math.GM], 2021.

%F From _Vaclav Kotesovec_, Dec 01 2016: (Start)

%F a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = A278300 = 2.455407482284127949... and c = 1.41510164826...

%F a(n) ~ c * d^n * n^(2*n + 1/2), where d = A278300/exp(2) = 0.332303267076220516... and c = 8.89134588451...

%F (End)

%p with(LinearAlgebra):

%p a:= n-> `if`(n=0, 1, Permanent(Matrix(n, (i, j)-> i+j))):

%p seq(a(n), n=0..16); # _Alois P. Heinz_, Nov 14 2016

%t f[i_, j_] := i + j;

%t m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]

%t TableForm[m[8]] (* 8x8 principal submatrix *)

%t Flatten[Table[f[i, n + 1 - i],

%t {n, 1, 12}, {i, 1, n}]] (* A003057 *)

%t Permanent[m_] :=

%t With[{a = Array[x, Length[m]]},

%t Coefficient[Times @@ (m.a), Times @@ a]];

%t Table[Permanent[m[n]], {n, 1, 15}] (* A204249 *)

%o (PARI) {a(n) = matpermanent(matrix(n, n, i, j, i+j))}

%o for(n=0, 20, print1(a(n), ", ")) \\ _Vaclav Kotesovec_, Dec 21 2018

%Y Cf. A000442, A003057, A085750, A085807, A278845, A278847, A278300, A278925, A278926.

%K nonn

%O 0,2

%A _Clark Kimberling_, Jan 14 2012

%E a(0)=1 prepended and one more term added by _Alois P. Heinz_, Nov 14 2016