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A356482
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a(n) is the hafnian of a symmetric Toeplitz matrix M(2*n) whose first row consists of 2*n, 2*n-1, ..., 1.
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7
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OFFSET
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0,3
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LINKS
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Table of n, a(n) for n=0..5.
Wikipedia, Hafnian
Wikipedia, Symmetric matrix
Wikipedia, Toeplitz Matrix
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EXAMPLE
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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.
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MATHEMATICA
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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]
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CROSSREFS
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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
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KEYWORD
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nonn,hard,more
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AUTHOR
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Stefano Spezia, Aug 09 2022
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STATUS
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approved
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