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A356491
a(n) is the permanent of a symmetric Toeplitz matrix M(n) whose first row consists of prime(1), prime(2), ..., prime(n).
3
1, 2, 13, 184, 4745, 215442, 14998965, 1522204560, 208682406913, 37467772675962, 8809394996942597, 2597094620811897948, 954601857873086235553, 428809643170145564168434, 229499307540038336275308821, 144367721963876506217872778284, 106064861375232790889279725340713
OFFSET
0,2
COMMENTS
Conjecture: a(n) is prime only for n = 1 and 2.
FORMULA
A351021(n) <= a(n) <= A351022(n).
EXAMPLE
For n = 1 the matrix M(1) is
2
with permanent a(1) = 2.
For n = 2 the matrix M(2) is
2, 3
3, 2
with permanent a(2) = 13.
For n = 3 the matrix M(3) is
2, 3, 5
3, 2, 3
5, 3, 2
with permanent a(3) = 184.
MAPLE
A356491 := proc(n)
local c ;
if n =0 then
return 1 ;
end if;
LinearAlgebra[ToeplitzMatrix]([seq(ithprime(c), c=1..n)], n, symmetric) ;
LinearAlgebra[Permanent](%) ;
end proc:
seq(A356491(n), n=0..15) ; # R. J. Mathar, Jan 31 2023
MATHEMATICA
k[i_]:=Prime[i]; M[ n_]:=ToeplitzMatrix[Array[k, n]]; a[n_]:=Permanent[M[n]]; Join[{1}, Table[a[n], {n, 16}]]
PROG
(PARI) a(n) = matpermanent(apply(prime, matrix(n, n, i, j, abs(i-j)+1))); \\ Michel Marcus, Aug 12 2022
(Python)
from sympy import Matrix, prime
def A356491(n): return Matrix(n, n, [prime(abs(i-j)+1) for i in range(n) for j in range(n)]).per() if n else 1 # Chai Wah Wu, Aug 12 2022
CROSSREFS
Cf. A005843 (trace of the matrix M(n)), A309131 (k-superdiagonal sum of the matrix M(n)), A356483 (hafnian of the matrix M(2*n)), A356490 (determinant of the matrix M(n)).
Sequence in context: A137610 A073178 A193192 * A226865 A062156 A371318
KEYWORD
nonn
AUTHOR
Stefano Spezia, Aug 09 2022
STATUS
approved