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A357279
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a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i, j] = i + j - 1.
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2
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1, 2, 43, 2610, 312081, 61825050, 18318396195, 7586241152490, 4184711271725985, 2965919152834367730, 2626408950849351178875
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OFFSET
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0,2
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COMMENTS
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The n X n matrix M is the n-th principal submatrix of A002024 considered as an array, and it is singular for n > 2.
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LINKS
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EXAMPLE
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a(2) = 43 because the hafnian of
1 2 3 4
2 3 4 5
3 4 5 6
4 5 6 7
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 43.
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MATHEMATICA
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M[i_, j_, n_]:=Part[Part[Table[r+c-1, {r, n}, {c, 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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PROG
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(PARI) tm(n) = matrix(n, n, i, j, i+j-1);
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
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CROSSREFS
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Cf. A002024, A002415 (absolute value of the coefficient of x^(n-2) in the characteristic polynomial of M(n)), A095833 (k-th super- and subdiagonal sums of the matrix M(n)), A204248 (permanent of M(n)).
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KEYWORD
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nonn,hard,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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