OFFSET
0,9
COMMENTS
This is the m=2, t=3 member of a two-parameter family of triangles such that T(n,k) is the number of tilings of an (n+(t-1)*k) X 1 board using k (1,m-1;t)-combs and n-k unit square tiles. A (1,g;t)-comb is composed of a line of t unit square tiles separated from each other by gaps of width g.
T(2*j+r-2*k,k) is the coefficient of x^k in (f(j,x))^(2-r)*(f(j+1,x))^r for r=0,1, where f(n,x) is a Narayana's cows polynomial defined by f(n,x)=f(n-1,x)+x*f(n-3,x)+delta(n,0) where f(n<0,x)=0.
T(n+4-2*k,k) is the number of subsets of {1,2,...,n} of size k such that no two elements in a subset differ by 2 or 4.
LINKS
Michael A. Allen, On a Two-Parameter Family of Generalizations of Pascal's Triangle, arXiv:2209.01377 [math.CO], 2022.
Michael A. Allen, On A Two-Parameter Family of Generalizations of Pascal's Triangle, J. Int. Seq. 25 (2022) Article 22.9.8.
FORMULA
T(n,0) = 1.
T(n,n) = delta(n mod 2,0).
T(n,1) = n-2 for n>1.
T(2*j-r,2*j-1) = 0 for j>0, r=0,1.
T(2*(j-1)+p,2*(j-1)) = j^p for j>0 and p=0,1,2.
T(2*(j-1)+3,2*(j-1)) = j^2*(j+1)/2 for j>0.
T(2*j+p,2*j-p) = C(j+1,2)^p for j>0 and p=0,1,2.
G.f. of row sums: (1-x)/(1-2*x).
G.f. of sums of T(n-2*k,k) over k: (1-x^3)/((1-x-x^3)*(1+x^4-x^6)).
T(n,k) = T(n-1,k) + T(n-1,k-1) for n>=2*k+1 if k>=0.
EXAMPLE
Triangle begins:
1;
1, 0;
1, 0, 1;
1, 1, 2, 0;
1, 2, 4, 0, 1;
1, 3, 6, 3, 3, 0;
1, 4, 9, 8, 9, 0, 1;
1, 5, 13, 17, 18, 6, 4, 0;
1, 6, 18, 30, 36, 20, 16, 0, 1;
1, 7, 24, 48, 66, 55, 40, 10, 5, 0;
1, 8, 31, 72, 114, 120, 100, 40, 25, 0, 1;
1, 9, 39, 103, 186, 234, 221, 135, 75, 15, 6, 0;
...
MATHEMATICA
T[n_, k_]:=If[k<0 || n<k, 0, T[n-1, k] + T[n-1, k-1] - T[n-2, k-1] + T[n-2, k-2] + T[n-3, k-1] - T[n-3, k-3] + KroneckerDelta[n, k, 0] - KroneckerDelta[n, k, 1]]; Table[T[n, k], {n, 0, 11}, {k, 0, n}]//Flatten
CROSSREFS
KEYWORD
AUTHOR
Michael A. Allen, Jun 04 2022
STATUS
approved