A391640 a(n) is the hafnian of the 2n X 2n Hankel matrix whose generic element is given by A000045(i+j-1) with i,j = 1, ..., n.

1, 1, 27, 7560, 20096706, 469234550400, 91647719244727380, 144882476551619079125580, 1810530857631422460202014819060, 175700644017711682078282449152774420640, 130588994626958481587169008526902476575988659000, 735177542920409283519725843348729064179058776531183808000

Offset

0,3

Comments

The Hankel transform of A000045 has the following polynomial as g.f. 1 + x + x^2: the matrices are singular for n > 2.

Links

Table of n, a(n) for n=0..11.

Wikipedia, Hafnian.

Wikipedia, Hankel matrix.

Wikipedia, Symmetric matrix.

Example

a(2) = 27 because the hafnian of
  [1, 1, 2,  3]
  [1, 2, 3,  5]
  [2, 3, 5,  8]
  [3, 5, 8, 13]
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 27.

Mathematica

M[i_, j_, n_]:=Part[Part[Table[Fibonacci[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]

Prog

(PARI) tm(n) = matrix(n, n, i, j, fibonacci(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, Mar 15 2026

Crossrefs

Cf. A000045, A120297, A357279, A374668, A393240.

Sequence in context: A013828 A394659 A343922 * A209261 A001321 A385211

Adjacent sequences: A391637 A391638 A391639 * A391641 A391642 A391643

Keyword

nonn,hard

Author

Stefano Spezia, Mar 15 2026

Extensions

a(6) from Michel Marcus, Mar 15 2026

a(7)-a(10) from Sean A. Irvine, Mar 18 2026

a(11) from Pontus von Brömssen, Mar 19 2026

Status

approved