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%I #15 Nov 22 2025 16:37:11
%S 1,1,4,9,27,59,147,309,680,1359,2745,5223,9923,18086,32732,57515,
%T 100133,170449,287377,475964,780971,1262965,2024448,3205891,5035293,
%U 7826984,12074562,18460301,28026831,42215156,63178973,93888989,138702881,203613174,297270739,431527023
%N Number of edge-weighted n-regular multigraphs on 2 unlabeled vertices with total edge weight n.
%H Andrew Howroyd, <a href="/A390166/b390166.txt">Table of n, a(n) for n = 0..500</a>
%e The a(3) = 9 multigraphs consist of either 3 edges connecting the two vertices or one edge between the vertices and a loop at each vertex. When there are 3 edges, possible edge weights are (3,0,0), (2,1,0) or (1,1,1). Otherwise with one edge and two loops possible weights are (3,0,0), (2,1,0), (1,2,0), (1,1,1), (0,3,0), (0,2,1), where (x,y,z) means the edge has weight x and the two loops weights y and z.
%o (PARI) vector(36,n,T(n-1,n-1)) \\ Needs T(n,k) from A390167.
%Y Main diagonal of A390167.
%K nonn
%O 0,3
%A _Andrew Howroyd_, Nov 22 2025