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

1, 2, 17, 336, 12052, 685080, 56658660, 6428352000, 958532774976, 181800011433600, 42745508545320000, 12203347213269273600, 4158410247782904833280, 1667267950805177583582720, 776990110000329481864608000, 416483579190482716042690560000

Offset: 0

Author

Clark Kimberling, Jan 14 2012

Keywords

• nonn

Comments

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

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..36
• Zhi-Wei Sun, Arithmetic properties of some permanents, arXiv:2108.07723 [math.GM], 2021.

Crossrefs

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

Programs

Maple

with(LinearAlgebra):
a:= n-> `if`(n=0, 1, Permanent(Matrix(n, (i, j)-> i+j))):
seq(a(n), n=0..16);  # Alois P. Heinz, Nov 14 2016

Mathematica

f[i_, j_] := i + j;
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 12}, {i, 1, n}]]  (* A003057 *)
Permanent[m_] :=
  With[{a = Array[x, Length[m]]},
   Coefficient[Times @@ (m.a), Times @@ a]];
Table[Permanent[m[n]], {n, 1, 15}]  (* A204249 *)

PARI

{a(n) = matpermanent(matrix(n, n, i, j, i+j))}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Dec 21 2018

Formula

From Vaclav Kotesovec, Dec 01 2016: (Start)

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

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

(End)

Extensions

a(0)=1 prepended and one more term added by Alois P. Heinz, Nov 14 2016