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).

1, 1, 17, 4929, 23872137, 1901611778409, 2469317979267366913, 52019468048773355156225921, 17726418489020770628047341494927089, 97518325438289444681986165275143492027985129, 8648473129650550498122567373327602114148485950241817345

Offset: 0

Author

Stefano Spezia, Sep 27 2022

Keywords

• nonn
• hard

Examples

			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.
		

Links

• Eric Weisstein's World of Mathematics, Pascal Matrix
• Wikipedia, Hafnian
• Wikipedia, Pascal matrix
• Wikipedia, Symmetric matrix

Crossrefs

Cf. A007318.

Cf. A006134 (trace of M(n)), A095833 (k-th super- and subdiagonal sums of M(n)), A320845 (permanent of M(n)).

Cf. A202038, A336114, A336286, A336400, A338456.

Cf. A356481, A356482, A356483, A356484.

Programs

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]

Extensions

a(6)-a(10) from Pontus von Brömssen, Oct 14 2023