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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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A356484 a(n) is the hafnian of a symmetric Toeplitz matrix M(2*n) whose first row consists of prime(2*n), prime(2*n-1), ..., prime(1).

Original entry on oeis.org

1, 2, 44, 5210, 1368900, 604109562, 535920536336, 728155179271474, 1103827431509790216, 2651375713654260218986, 7537958658258053003685636
Offset: 0

Views

Author

Stefano Spezia, Aug 09 2022

Keywords

Comments

a(n) is even for n >= 1. - Robert Israel, Oct 13 2023

Examples

			a(2) = 44 because the hafnian of
    7  5  3  2
    5  7  5  3
    3  5  7  5
    2  3  5  7
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 44.

Links

Crossrefs

Cf. A202038, A336114, A336286, A336400, A338456.
Cf. A356481, A356482, A356483.
Cf. A356492 (determinant of M(n)), A356493 (permanent of M(n)).

Programs

  • Maple
    haf:= proc(A)
    local n, s, Pairpart, p;
    Pairpart := proc(L) local j, t; if L = {} then return {{}}; end if; {seq(seq({{L[1], L[j]}} union t, t = procname(L minus {L[1], L[j]})), j = 2 .. nops(L))}; end proc;
    n := LinearAlgebra:-Dimension(A);
    if n[1] <> n[2] then
        error "must be square matrix";
    end if;
    n := n[1];
    if n::odd then
        error "dimension of matrix must be even";
    end if;
    add(mul(A[s[1], s[2]], s = p), p = Pairpart({$ (1 .. n)}));
end proc:
f:= proc(n) local i; haf(LinearAlgebra:-ToeplitzMatrix([seq(ithprime(i),i=2*n..1,-1)],symmetric)) end proc:
f(0):= 1:
map(f, [$0..7]); # Robert Israel, Oct 13 2023
  • Mathematica
    k[i_]:=Prime[i]; M[i_, j_, n_]:=Part[Part[ToeplitzMatrix[Reverse[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(n-j+1), if (j==1, prime(n-i+1))))); 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(9) from Robert Israel, Oct 13 2023
a(10) from Pontus von Brömssen, Oct 14 2023

A356493 a(n) is the permanent of a symmetric Toeplitz matrix M(n) whose first row consists of prime(n), prime(n-1), ..., prime(1).

Original entry on oeis.org

1, 2, 13, 271, 12030, 1346758, 214022024, 51763672608, 16088934953136, 6611717516842608, 4412314619046451200, 3533754988232088933120, 3506189715435673999194112, 4444138735439968822425464576, 5893766827264238066914528545792, 8502284313901016361834901076874240, 15350799440394462109333953415858960384
Offset: 0

Views

Author

Stefano Spezia, Aug 09 2022

Keywords

Comments

Conjecture: a(n) is prime only for n = 1, 2, and 3.
Conjecture is true because a(n) is even for n >= 4. This is because a(n) == A356492(n) (mod 2), and all but two rows of the matrix consist of odd numbers. - Robert Israel, Oct 13 2023

Examples

			For n = 1 the matrix M(1) is
    2
with permanent a(1) = 2.
For n = 2 the matrix M(2) is
    3, 2
    2, 3
with permanent a(2) = 13.
For n = 3 the matrix M(3) is
    5, 3, 2
    3, 5, 3
    2, 3, 5
with permanent a(3) = 271.

Links

Crossrefs

Cf. A033286 (trace of the matrix M(n)), A356484 (hafnian of the matrix M(2*n)), A356492 (determinant of the matrix M(n)).
Cf. A351021, A351022.

Programs

  • Mathematica
    k[i_]:=Prime[i]; M[ n_]:=ToeplitzMatrix[Reverse[Array[k, n]]]; a[n_]:=Permanent[M[n]]; PrimeQ[Join[{1},Table[a[n],{n,16}]]]
  • PARI
    a(n) = matpermanent(apply(prime, matrix(n,n,i,j,n-abs(i-j)))); \\ Michel Marcus, Aug 12 2022

Formula

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

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