A354668 Triangle read by rows: T(n,k) is the number of tilings of an (n+2*k) X 1 board using k (1,2;3)-combs and n-k squares.

1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 2, 0, 1, 1, 3, 4, 0, 0, 1, 2, 5, 8, 0, 0, 1, 1, 3, 8, 12, 0, 3, 3, 0, 1, 4, 12, 18, 9, 12, 9, 0, 0, 1, 5, 16, 27, 25, 29, 27, 0, 0, 1, 1, 6, 21, 42, 51, 66, 54, 0, 6, 4, 0, 1, 7, 27, 62, 95, 135, 108, 36

Offset: 0

Author

Michael A. Allen, Jul 30 2022

Keywords

• easy
• nonn
• tabl

Comments

This is the m=3, 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(3*j+r-2*k,k) is the coefficient of x^k in (f(j,x))^(3-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+6-2*k,k) is the number of subsets of {1,2,...,n} of size k such that no two elements in a subset differ by 3 or 6.

Examples

			Triangle begins:
  1;
  1,   0;
  1,   0,   0;
  1,   0,   0,   1;
  1,   0,   1,   2,   0;
  1,   1,   3,   4,   0,   0;
  1,   2,   5,   8,   0,   0,   1;
  1,   3,   8,  12,   0,   3,   3,   0;
  1,   4,  12,  18,   9,  12,   9,   0,   0;
  1,   5,  16,  27,  25,  29,  27,   0,   0,   1;
  1,   6,  21,  42,  51,  66,  54,   0,   6,   4,   0;
  1,   7,  27,  62,  95, 135, 108,  36,  30,  16,   0,   0;
...
		

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.

Crossrefs

Sums over k of T(n-2*k,k) are A224810.

Other members of the family of triangles: A007318 (m=1,t=2), A059259 (m=2,t=2), A350110 (m=3,t=2), A350111 (m=4,t=2), A350112 (m=5,t=2), A354665 (m=2,t=3), A354666 (m=2,t=4), A354667 (m=2,t=5).

Other triangles related to tiling using combs: A059259, A123521, A157897, A335964.

Programs

Mathematica

f[n_]:=If[n<0, 0, f[n-1]+x*f[n-3]+KroneckerDelta[n,0]]; T[n_, k_]:=Module[{j=Floor[(n+2*k)/3], r=Mod[n+2*k,3]}, Coefficient[f[j]^(3-r)*f[j+1]^r, x, k]]; Flatten@Table[T[n,k], {n, 0, 11}, {k, 0, n}]

Formula

T(n,0) = 1.

T(n,n) = delta(n mod 3,0).

T(n,1) = n-4 for n>3.

T(3*j-r,3*j-p) = 0 for j>0, p=1,2, and r=1-p,...,p.

T(n,2*j) = C(n/2,j)^2 for j>0 and n even and 2*j <= n <= 2*j+8.

T(n,2*j) = C((n-1)/2,j)*C((n+1)/2,j) for j>0 and n odd and 2*j < n < 2*j+8.

T(2*j+3*p,2*j-p) = C(j+3,4)^p for j>0 and p=0,1,2.

G.f. of sums of T(n-2*k,k) over k: (1+x^3-x^4-x^5+x^6-2*x^7-x^8-x^9-2*x^10-x^12-x^13-x^15)/((1-x)*(1+x+x^2)*(1-x-x^3)*(1+3*x^3+7*x^6+9*x^9+7*x^12+3*x^15+x^18)).

T(n,k) = T(n-1,k) + T(n-1,k-1) for n>=4*k+1 if k>=0.