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

0,12

Comments

Row sums are A000930.

References

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

Links

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

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

Example

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;

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

Crossrefs

Cf. A046854.

Sequence in context: A080576 A321744 A322763 * A294775 A369929 A330461

Adjacent sequences: A213208 A213209 A213210 * A213212 A213213 A213214

Keyword

nonn,tabl

Author

Geoffrey Critzer, Mar 02 2013

Status

approved