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).

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

Offset

0,2

Comments

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

Links

Table of n, a(n) for n=0..10.

Wikipedia, Hafnian

Wikipedia, Symmetric matrix

Wikipedia, Toeplitz Matrix

Example

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.

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]

Prog

(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

Crossrefs

Cf. A202038, A336114, A336286, A336400, A338456.

Cf. A356481, A356482, A356483.

Cf. A356492 (determinant of M(n)), A356493 (permanent of M(n)).

Sequence in context: A054914 A329021 A275307 * A291808 A161745 A048566

Adjacent sequences: A356481 A356482 A356483 * A356485 A356486 A356487

Keyword

nonn,hard,more

Author

Stefano Spezia, Aug 09 2022

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

Status

approved