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A213211 Triangular array read by rows: T(n,k) is the number of size k subsets of {1,2,...,n} such that (when the elements are arranged in increasing order) the smallest element is congruent to 1 mod 3 and the difference of every pair of successive elements is also congruent to 1 mod 3.

%I A213211 #14 Mar 12 2013 09:50:48
%S A213211 1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,3,1,1,1,1,2,3,4,1,1,1,1,3,3,4,5,1,
%T A213211 1,1,1,3,6,4,5,6,1,1,1,1,3,6,10,5,6,7,1,1,1,1,4,6,10,15,6,7,8,1,1,1,1,
%U A213211 4,10,10,15,21,7,8,9,1,1,1,1,4,10,20,15,21,28,8,9,10,1,1,1
%N A213211 Triangular array read by rows: T(n,k) is the number of size k subsets of {1,2,...,n} such that (when the elements are arranged in increasing order) the smallest element is congruent to 1 mod 3 and the difference of every pair of successive elements is also congruent to 1 mod 3.
%C A213211 Row sums are A000930.
%D A213211 Combinatorial Enumeration, I. Goulden and D. Jackson, John Wiley and Sons, 1983, page 56.
%H A213211 Alois P. Heinz, <a href="/A213211/b213211.txt">Rows n = 0..140, flattened</a>
%F A213211 G.f.: (1 + x + x^2)/(1 - x^3 - y*x).
%F A213211 T(n,k) = C(k+floor((n-k)/3),k). - _Alois P. Heinz_, Mar 02 2013
%e A213211 T(6,3) = 4 because we have: {1,2,3}, {1,2,6}, {1,5,6}, {4,5,6}.
%e A213211 1;
%e A213211 1, 1;
%e A213211 1, 1, 1;
%e A213211 1, 1, 1,  1;
%e A213211 1, 2, 1,  1,  1;
%e A213211 1, 2, 3,  1,  1, 1;
%e A213211 1, 2, 3,  4,  1, 1, 1;
%e A213211 1, 3, 3,  4,  5, 1, 1, 1;
%e A213211 1, 3, 6,  4,  5, 6, 1, 1, 1;
%e A213211 1, 3, 6, 10,  5, 6, 7, 1, 1, 1;
%e A213211 1, 4, 6, 10, 15, 6, 7, 8, 1, 1, 1;
%p A213211 T:= (n, k)-> binomial(k+floor((n-k)/3), k):
%p A213211 seq(seq(T(n,k), k=0..n), n=0..14);  # _Alois P. Heinz_, Mar 02 2013
%t A213211 nn=10;f[list_]:=Select[list,#>0&];Map[f,CoefficientList[Series[ (1+x+x^2)/(1-x^3-y x),{x,0,nn}],{x,y}]]//Grid
%Y A213211 Cf. A046854.
%K A213211 nonn,tabl
%O A213211 0,12
%A A213211 _Geoffrey Critzer_, Mar 02 2013