A322363 Permanent of the matrix [i^(j-1)]_{i,j=1..n}.

1, 3, 48, 6160, 8527540, 159676348608, 48587473295391744, 280486193644638162542592, 35123514116515156931718809604096, 107372634058167343575121983395332766269440, 8905983201619001018383658118740652467256553624043520, 22052482744180702505678193326985890418061231090612350123937628160

Offset

1,2

Comments

Conjecture:

(i) n+1 | a(n) if and only if n is not congruent to 1 modulo 4.

(ii) For any Fermat prime p, we have a(p-1) == ((p-1)/2)!*p (mod p^2). If n > 1 is neither congruent to 2 modulo 4 nor a Fermat prime, then a(n-1) == 0 (mod n^2).

The author has proved that p | a(p-1) for any odd prime p. This implies that n | a(n) for all n > 2.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..33 (terms 1..15 from Zhi-Wei Sun, terms 16..30 from Jinyuan Wang)

Zhi-Wei Sun, On the permanent per[i^(j-1)]_{i,j=1..n} modulo p^2, Question 316839 on Mathoverflow, Dec. 3, 2018.

Zhi-Wei Sun, On restricted permutations of {1,...,n}, arXiv:1811.10503 [math.CO], 2018.

Example

a(2) = 3 since permanent[i^(j-1)]_{i,j=1,2} = 1*2 + 1*1 = 3.

Mathematica

Permanent[m_List]:=With[{v = Array[x, Length[m]]}, Coefficient[Times @@ (m.v), Times @@ v]];
a[n_]:=a[n]=Permanent[Table[i^(j-1), {i, 1, n}, {j, 1, n}]];
Do[Print[n, " ", a[n]], {n, 1, 15}]

Prog

(PARI) a(n) = matpermanent(matrix(n, n, i, j, i^(j-1))); \\ Michel Marcus, Dec 05 2018

Crossrefs

Cf. A000040, A019434.

Sequence in context: A003029 A049524 A270748 * A359965 A033493 A334421

Adjacent sequences: A322360 A322361 A322362 * A322364 A322365 A322366

Keyword

nonn

Author

Zhi-Wei Sun, Dec 04 2018

Status

approved