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%I #10 Mar 14 2021 00:47:01
%S 1,4,32,370,5950,122984,3175696,98815588,3638940860,155377163440,
%T 7598445388096,420034502219864,26014375783223272,1788772035008337760,
%U 135644687161742899520,11268192704027639350384,1020100484786824631520016,100126060947226759050509888
%N Number of matchings in the 3 X n rook graph.
%H Andrew Howroyd, <a href="/A341502/b341502.txt">Table of n, a(n) for n = 0..100</a>
%F a(n) = Sum{i,j,k>=0, i+j+k<=n} n!/(i!*j!*k!*(n-i-j-k)!) * A000085(n-i-j) * A000085(n-i-k) * A000085(n-j-k).
%o (PARI) \\ here b(n) is A000085.
%o b(n)={sum(k=0, n\2, n!/((n-2*k)!*2^k*k!))}
%o a(n)={my(v=vector(n+1, i, b(i-1))); sum(i=0,n, sum(j=0, n-i, sum(k=0, n-i-j, n!/(i!*j!*k!*(n-i-j-k)!)*v[1+n-i-j]*v[1+n-i-k]*v[1+n-j-k] )))}
%Y Row 3 of A270227.
%Y Cf. A000085, A341848, A341851.
%K nonn
%O 0,2
%A _Andrew Howroyd_, Feb 21 2021
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