A354666 Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k-1) + 2*T(n-2,k-2) - T(n-3,k-1) - T(n-3,k-2) + T(n-4,k-1) + T(n-4,k-2) - T(n-4,k-3) - T(n-4,k-4) + delta(n,0)*delta(k,0) - delta(n,2)*(delta(k,1) + delta(k,2)), T(n

1, 1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 1, 4, 0, 1, 1, 2, 6, 0, 3, 0, 1, 3, 9, 4, 9, 0, 1, 1, 4, 12, 10, 18, 0, 4, 0, 1, 5, 16, 21, 36, 10, 16, 0, 1, 1, 6, 21, 36, 60, 30, 40, 0, 5, 0, 1, 7, 27, 57, 100, 81, 100, 20, 25, 0, 1, 1, 8, 34, 84, 158, 168

Offset: 0

Author

Michael A. Allen, Jun 04 2022

Keywords

• easy
• nonn
• tabl

Comments

This is the m=2, t=4 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-3*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 (1,4)-bonacci polynomial defined by f(n,x)=f(n-1,x)+x*f(n-4,x)+delta(n,0) where f(n<0,x)=0.

T(n+6-3*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, 4, or 6.

Examples

			Triangle begins:
  1;
  1,   0;
  1,   0,   1;
  1,   0,   2,   0;
  1,   1,   4,   0,   1;
  1,   2,   6,   0,   3,   0;
  1,   3,   9,   4,   9,   0,   1;
  1,   4,  12,  10,  18,   0,   4,   0;
  1,   5,  16,  21,  36,  10,  16,   0,   1;
  1,   6,  21,  36,  60,  30,  40,   0,   5,   0;
  1,   7,  27,  57, 100,  81, 100,  20,  25,   0,   1;
  1,   8,  34,  84, 158, 168, 200,  70,  75,   0,   6,   0;
  1,   9,  42, 118, 243, 322, 400, 231, 225,  35,  36,   0,   1;
...
		

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

Row sums are A099163.

Sums over k of T(n-3*k,k) are A224808.

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), A354667 (m=2,t=5), A354668 (m=3,t=3).

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

Programs

Mathematica

T[n_,k_]:=If[k<0 || n
				

Formula

T(n,0) = 1.

T(n,n) = delta(n mod 2,0).

T(n,1) = n-3 for n>2.

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

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

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

T(2*j+3,2*(j-1)) = (j*(j+1))^2*(j+2)/12 for j>0.

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

G.f. of row sums: (1-2*x^2)/(1-x-3*x^2+2*x^3).

G.f. of sums of T(n-3*k,k) over k: (1-x^5-x^8)/(1-x-x^5+x^6-x^7-2*x^8+x^9-x^10+x^13+x^16).

T(n,k) = T(n-1,k) + T(n-1,k-1) for n>=3*k+1 if k>=0.