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.

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, 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, 4, 10, 10, 15, 21, 7, 8, 9, 1, 1, 1, 1, 4, 10, 20, 15, 21, 28, 8, 9, 10, 1, 1, 1

Offset: 0

Author

Geoffrey Critzer, Mar 02 2013

Keywords

• nonn
• tabl

Comments

Row sums are A000930.

Examples

			T(6,3) = 4 because we have: {1,2,3}, {1,2,6}, {1,5,6}, {4,5,6}.
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, 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;
		

References

• Combinatorial Enumeration, I. Goulden and D. Jackson, John Wiley and Sons, 1983, page 56.

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Cf. A046854.

Programs

Maple

T:= (n, k)-> binomial(k+floor((n-k)/3), k):
seq(seq(T(n,k), k=0..n), n=0..14);  # Alois P. Heinz, Mar 02 2013

Mathematica

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

Formula

G.f.: (1 + x + x^2)/(1 - x^3 - y*x).

T(n,k) = C(k+floor((n-k)/3),k). - Alois P. Heinz, Mar 02 2013