This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A357419 #16 Feb 16 2025 08:34:04
%S A357419 1,1,17,4929,23872137,1901611778409,2469317979267366913,
%T A357419 52019468048773355156225921,17726418489020770628047341494927089,
%U A357419 97518325438289444681986165275143492027985129,8648473129650550498122567373327602114148485950241817345
%N 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).
%H A357419 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PascalMatrix.html">Pascal Matrix</a>
%H A357419 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hafnian">Hafnian</a>
%H A357419 Wikipedia, <a href="https://en.wikipedia.org/wiki/Pascal_matrix">Pascal matrix</a>
%H A357419 Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_matrix">Symmetric matrix</a>
%e A357419 a(2) = 17 because the hafnian of
%e A357419 1, 1, 1, 1
%e A357419 1, 2, 3, 4
%e A357419 1, 3, 6, 10
%e A357419 1, 4, 10, 20
%e A357419 equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 17.
%t A357419 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]
%Y A357419 Cf. A007318.
%Y A357419 Cf. A006134 (trace of M(n)), A095833 (k-th super- and subdiagonal sums of M(n)), A320845 (permanent of M(n)).
%Y A357419 Cf. A202038, A336114, A336286, A336400, A338456.
%Y A357419 Cf. A356481, A356482, A356483, A356484.
%K A357419 nonn,hard
%O A357419 0,3
%A A357419 _Stefano Spezia_, Sep 27 2022
%E A357419 a(6)-a(10) from _Pontus von Brömssen_, Oct 14 2023