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A357419
a(n) is the hafnian of the 2n X 2n symmetric Pascal matrix defined by M[i, j] = A007318(i + j - 2, i - 1).
0
1, 1, 17, 4929, 23872137, 1901611778409, 2469317979267366913, 52019468048773355156225921, 17726418489020770628047341494927089, 97518325438289444681986165275143492027985129, 8648473129650550498122567373327602114148485950241817345
OFFSET
0,3
LINKS
Eric Weisstein's World of Mathematics, Pascal Matrix
Wikipedia, Hafnian
Wikipedia, Pascal matrix
Wikipedia, Symmetric matrix
EXAMPLE
a(2) = 17 because the hafnian of
1, 1, 1, 1
1, 2, 3, 4
1, 3, 6, 10
1, 4, 10, 20
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 17.
MATHEMATICA
M[i_, j_, n_]:=Part[Part[Table[Binomial[r+c-2, r-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]
CROSSREFS
Cf. A007318.
Cf. A006134 (trace of M(n)), A095833 (k-th super- and subdiagonal sums of M(n)), A320845 (permanent of M(n)).
Sequence in context: A350980 A161583 A013722 * A238610 A052286 A079687
KEYWORD
nonn,hard
AUTHOR
Stefano Spezia, Sep 27 2022
EXTENSIONS
a(6)-a(10) from Pontus von Brömssen, Oct 14 2023
STATUS
approved