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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). 9
1, 2, 44, 5210, 1368900, 604109562, 535920536336, 728155179271474, 1103827431509790216, 2651375713654260218986, 7537958658258053003685636 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
a(n) is even for n >= 1. - Robert Israel, Oct 13 2023
LINKS
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. A356492 (determinant of M(n)), A356493 (permanent of M(n)).
Sequence in context: A054914 A329021 A275307 * A291808 A161745 A048566
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

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Last modified April 18 11:52 EDT 2024. Contains 371779 sequences. (Running on oeis4.)