cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A356483 a(n) is the hafnian of a symmetric Toeplitz matrix M(2*n) whose first row consists of prime(1), prime(2), ..., prime(2*n).

Original entry on oeis.org

1, 3, 55, 2999, 347391, 69702479, 22441691645, 10776262328919, 7190279422736061, 6439969796874334809, 7447188585071730451961
Offset: 0

Views

Author

Stefano Spezia, Aug 09 2022

Keywords

Examples

			a(2) = 55 because the hafnian of
    2  3  5  7
    3  2  3  5
    5  3  2  3
    7  5  3  2
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 55.

Links

Crossrefs

Cf. A202038, A336114, A336286, A336400, A338456.
Cf. A356481, A356482, A356484.
Cf. A356490 (determinant of M(n)), A356491 (permanent of M(n)).

Programs

  • Mathematica
    k[i_]:=Prime[i]; M[i_, j_, n_]:=Part[Part[ToeplitzMatrix[Array[k, n]], i], j]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i], 2n], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 6, 0]
  • PARI
    tm(n) = my(m = matrix(n, n, i, j, if (i==1, prime(j), if (j==1, prime(i))))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m;
a(n) = my(m = tm(2*n), s=0); forperm([1..2*n], p, s += prod(j=1, n, m[p[2*j-1], p[2*j]]); ); s/(n!*2^n); \\ Michel Marcus, May 02 2023

Extensions

a(6) from Michel Marcus, May 02 2023
a(7)-a(10) from Pontus von Brömssen, Oct 14 2023

A356491 a(n) is the permanent of a symmetric Toeplitz matrix M(n) whose first row consists of prime(1), prime(2), ..., prime(n).

Original entry on oeis.org

1, 2, 13, 184, 4745, 215442, 14998965, 1522204560, 208682406913, 37467772675962, 8809394996942597, 2597094620811897948, 954601857873086235553, 428809643170145564168434, 229499307540038336275308821, 144367721963876506217872778284, 106064861375232790889279725340713
Offset: 0

Views

Author

Stefano Spezia, Aug 09 2022

Keywords

Comments

Conjecture: a(n) is prime only for n = 1 and 2.

Examples

			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.

Links

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)).
Cf. A351021, A351022.

Programs

  • 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}]]
  • 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

Formula

A351021(n) <= a(n) <= A351022(n).
Showing 1-2 of 2 results.

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