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A356482 a(n) is the hafnian of a symmetric Toeplitz matrix M(2*n) whose first row consists of 2*n, 2*n-1, ..., 1. 7
1, 1, 16, 714, 62528, 9056720 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..5.

Wikipedia, Hafnian

Wikipedia, Symmetric matrix

Wikipedia, Toeplitz Matrix

EXAMPLE

a(2) = 16 because the hafnian of

4 3 2 1

3 4 3 2

2 3 4 3

1 2 3 4

equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 16.

MATHEMATICA

k[i_]:=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]

CROSSREFS

Cf. A001792 (determinant of M(n)), A307783.

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

Cf. A356481, A356483, A356484.

Sequence in context: A283534 A294704 A264114 * A201622 A220809 A221023

Adjacent sequences: A356479 A356480 A356481 * A356483 A356484 A356485

KEYWORD

nonn,hard,more

AUTHOR

Stefano Spezia, Aug 09 2022

STATUS

approved

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Last modified February 4 05:49 EST 2023. Contains 360046 sequences. (Running on oeis4.)