A278926 -id:A278926 - cp's OEIS frontend

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

Clark Kimberling, Jan 14 2012

nonn

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

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.

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

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

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 *)

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

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)

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

A278847 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = i^2 + j^2.

Original entry on oeis.org

1, 2, 41, 3176, 620964, 246796680, 174252885732, 199381727959680, 345875291854507584, 864860593764292790400, 2996169331694350840741440, 13929521390709644084719495680, 84659009841182126038701730464000, 658043094413184868424932006273344000

Offset: 0

Vaclav Kotesovec, Nov 29 2016

nonn

From Zhi-Wei Sun, Aug 19 2021: (Start)
I have proved that a(n) == (-1)^(n-1)*2*n! (mod 2n+1) whenever 2n+1 is prime.
Conjecture 1: If 2n+1 is composite, then a(n) == 0 (mod 2n+1).
Conjecture 2: If p = 4n+1 is prime, then the sum of those Product_{j=1..2n}(j^2-f(j)^2)^{-1} with f over all the derangements of {1,...,2n} is congruent to 1/(n!)^2 modulo p. (End)

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.

Cf. A005249, A085750, A085807, A204249, A278845, A278925, A278926, A346934, A346949.

with(LinearAlgebra):
a:= n-> `if`(n=0, 1, Permanent(Matrix(n, (i, j)-> i^2+j^2))):
seq(a(n), n=0..16);  # after Alois P. Heinz

Flatten[{1, Table[Permanent[Table[i^2+j^2, {i, 1, n}, {j, 1, n}]], {n, 1, 15}]}]

a(n)={matpermanent(matrix(n, n, i, j, i^2 + j^2))} \\ Andrew Howroyd, Aug 21 2018

a(n) ~ c * d^n * (n!)^3 / n, where d = 3.809076776112918119... and c = 1.07739642254738...

A278925 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = i^3 + j^3.

Original entry on oeis.org

1, 2, 113, 38736, 46311652, 143820883800, 966462062838180, 12412328008727861760, 278484670746890475310656, 10197331743850942940587152000, 577793817845799602600135280168000, 48534819511412868687827815575204633600, 5834998526939444017550860154062183732711680

Offset: 0

Vaclav Kotesovec, Dec 01 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..36

Cf. A204249, A278847, A278926.

with(LinearAlgebra):
a:= n-> `if`(n=0, 1, Permanent(Matrix(n, (i, j)-> i^3+j^3))):
seq(a(n), n=0..16);

Flatten[{1, Table[Permanent[Table[i^3+j^3, {i, 1, n}, {j, 1, n}]], {n, 1, 15}]}]

{a(n) = matpermanent(matrix(n, n, i, j, i^3+j^3))}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Dec 21 2018

a(n) ~ c * d^n * n!^4 / n^(3/2), where d = 6.538385468679... and c = 0.84959670006...

cp's OEIS Frontend

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

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A278847 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = i^2 + j^2.

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Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A278925 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = i^3 + j^3.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula