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%I #5 Mar 09 2020 21:18:38
%S 1,0,1,1,1,1,1,1,0,2,0,1,1,3,8,8,3,1,1,0,38,0,38,0,1,1,15,208,730,730,
%T 208,15,1,1,0,1348,0,20670,0,1348,0,1,1,105,10126,188790,781578,
%U 781578,188790,10126,105,1,1,0,86174,0,37885204,0,37885204,0,86174,0,1
%N Irregular triangle read by rows: T(n,k) is the number of k-regular graphs on n labeled nodes with loops allowed, n >= 1, 0 <= k <= n + 1.
%C A loop adds 2 to the degree of its vertex.
%F T(n,k) = T(n, n+1-k).
%e Triangle begins:
%e 1, 0, 1;
%e 1, 1, 1, 1;
%e 1, 0, 2, 0, 1;
%e 1, 3, 8, 8, 3, 1;
%e 1, 0, 38, 0, 38, 0, 1;
%e 1, 15, 208, 730, 730, 208, 15, 1;
%e 1, 0, 1348, 0, 20670, 0, 1348, 0, 1;
%e 1, 105, 10126, 188790, 781578, 781578, 188790, 10126, 105, 1;
%e ...
%Y Row sums are A322635.
%Y Columns k=0..4 are A000012, A123023, A108246, A110039 (with interspersed zeros), A228697.
%Y Cf. A059441, A333157.
%K nonn,tabf
%O 1,10
%A _Andrew Howroyd_, Mar 09 2020