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A338456 a(n) is the hafnian of a symmetric Toeplitz matrix M(2n) whose first row consists of a single zero followed by successive positive integers repeated (A004526). 1
1, 1, 4, 45, 968, 34265 (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) = 4 because the hafnian of

0  1  1  2

1  0  1  1

1  1  0  1

2  1  1  0

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

MATHEMATICA

k[i_]:=Floor[i/2]; A[i_, j_, n_]:=Part[Part[ToeplitzMatrix[Array[k, n], Array[k, n]], i], j]; a[n_]:=Sum[Product[A[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i], 2n], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 5, 0]

PROG

(PARI) tm(n) = {my(m = matrix(n, n, i, j, if (i==1, j\2, if (j==1, i\2)))); 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, Nov 11 2020

CROSSREFS

Cf. A004526.

Cf. A002378 (conjectured determinant of M(2n+1)), A083392 (conjectured determinant of M(n+1)), A332566 (permanent of M(n)), A333119 (k-th super- and subdiagonal sums of the matrix M(n)).

Cf. A202038, A336114, A336286, A336400.

Sequence in context: A107668 A214400 A197989 * A276292 A174484 A158887

Adjacent sequences:  A338453 A338454 A338455 * A338457 A338458 A338459

KEYWORD

nonn,hard,more

AUTHOR

Stefano Spezia, Oct 28 2020

EXTENSIONS

a(5) from Michel Marcus, Nov 11 2020

STATUS

approved

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Last modified January 26 06:41 EST 2022. Contains 350572 sequences. (Running on oeis4.)