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A336667 Triangular array read by rows. T(n,k) is the number of closed walks of length 2n along the edges of a cube based at vertex v that return to v exactly k times, n>=0, 0<=k<=n.

%I A336667 #12 Aug 03 2020 02:59:59
%S A336667 1,0,3,0,12,9,0,84,72,27,0,588,648,324,81,0,4116,5544,3564,1296,243,0,
%T A336667 28812,45864,35748,16848,4860,729,0,201684,370440,337932,193104,72900,
%U A336667 17496,2187
%N A336667 Triangular array read by rows. T(n,k) is the number of closed walks of length 2n along the edges of a cube based at vertex v that return to v exactly k times, n>=0, 0<=k<=n.
%H A336667 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; page 340.
%F A336667 O.g.f.: (1 - 7*x^2)/(1 - 7*x^2 - 3*y*x^2 + 9*y*x^4).
%e A336667 Triangle T(n,k) begins:
%e A336667   1;
%e A336667   0,   3;
%e A336667   0,  12,   9;
%e A336667   0,  84,  72,  27;
%e A336667   0, 588, 648, 324, 81;
%e A336667   ...
%t A336667 Table[nn = n; CoefficientList[Series[(1 - 7 z^2)/(1 - (7 + 3 u) z^2 + 9 u z^4), {z, 0, nn}], {z,u}][[-1]], {n, 0, 15, 2}] // Grid
%Y A336667 Cf. A054879 (row sums), A328778 (column k=1).
%K A336667 nonn,tabl,easy
%O A336667 0,3
%A A336667 _Geoffrey Critzer_, Jul 29 2020