A350110 -id:A350110 - cp's OEIS frontend

A350111 Triangle read by rows: T(n,k) is the number of tilings of an (n+k)-board using k (1,3)-fences and n-k squares.

1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 3, 4, 2, 0, 1, 3, 6, 7, 4, 0, 0, 1, 4, 9, 12, 8, 0, 0, 0, 1, 5, 13, 20, 16, 8, 4, 2, 1, 1, 6, 18, 32, 36, 28, 19, 12, 3, 0, 1, 7, 24, 50, 69, 69, 58, 31, 9, 0, 0, 1, 8, 31, 74, 120, 144, 127, 78, 27, 0, 0, 0

Offset: 0

Michael A. Allen, Dec 22 2021

This is the m=4 member in the sequence of triangles A007318, A059259, A350110, A350111, A350112 which give the number of tilings of an (n+k) X 1 board using k (1,m-1)-fences and n-k unit square tiles. A (1,g)-fence is composed of two unit square tiles separated by a gap of width g.
It is also the m=4, t=2 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(4*j+r-k,k) is the coefficient of x^k in (f(j,x))^(4-r)*(f(j+1,x))^r for r=0,1,2,3 where f(n,x) is one form of a Fibonacci polynomial defined by f(n+1,x)=f(n,x)+x*f(n-1,x) where f(0,x)=1 and f(n<0,x)=0.
T(n+4-k,k) is the number of subsets of {1,2,...,n} of size k such that no two elements in a subset differ by 4.
Sum of (n+3)-th antidiagonal (counting initial 1 as the 0th) is A031923(n).

			Triangle begins:
  1;
  1,   0;
  1,   0,   0;
  1,   0,   0,   0;
  1,   1,   1,   1,   1;
  1,   2,   3,   4,   2,   0;
  1,   3,   6,   7,   4,   0,   0;
  1,   4,   9,  12,   8,   0,   0,   0;
  1,   5,  13,  20,  16,   8,   4,   2,   1;
  1,   6,  18,  32,  36,  28,  19,  12,   3,   0;
  1,   7,  24,  50,  69,  69,  58,  31,   9,   0,   0;
  1,   8,  31,  74, 120, 144, 127,  78,  27,   0,   0,   0;
  1,   9,  39, 105, 195, 264, 265, 189,  81,  27,   9,   3,   1;
  1,  10,  48, 144, 300, 458, 522, 432, 270, 132,  58,  24,   4,   0;

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.
Michael A. Allen and Kenneth Edwards, On Two Families of Generalizations of Pascal's Triangle, J. Int. Seq. 25 (2022) Article 22.7.1.

Sums of antidiagonals: A031923
Other members of the two-parameter family of triangles: A007318 (m=1,t=2), A059259 (m=2,t=2), A350110 (m=3,t=2), A350112 (m=5,t=2), A354665 (m=2,t=3), A354666 (m=2,t=4), A354667 (m=2,t=5), A354668 (m=3,t=3).
Other triangles related to tiling using fences: A123521, A157897, A335964.

f[n_]:=If[n<0,0,f[n-1]+x*f[n-2]+KroneckerDelta[n,0]];
T[n_, k_]:=Module[{j=Floor[(n+k)/4],r=Mod[n+k,4]},
  Coefficient[f[j]^(4-r)*f[j+1]^r,x,k]];
Flatten@Table[T[n,k], {n, 0, 13}, {k, 0, n}]
(* or *)
T[n_,k_]:=If[k<0 || n

T(n,k) = T(n-1,k) + T(n-2,k-1) - T(n-3,k-1) + T(n-3,k-2) + T(n-4,k-1) + T(n-4,k-3) + 2*T(n-4,k-4) + T(n-5,k-2) + 2*T(n-5,k-3) - T(n-5,k-4) - T(n-6,k-3)-T(n-6,k-5) - T(n-7,k-4)-T(n-7,k-5) - T(n-7,k-6) - T(n-8,k-7)-T(n-8,k-8) + delta(n,0)*delta(k,0) - delta(n,2)*delta(k,1) - delta(n,3)*delta(k,2) - delta(n,4)*delta(k,4) with T(n

T(n,0) = 1.

T(n,n) = delta(n mod 4,0).

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

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

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

T(4*j+1,4*j-1) = 4*j(j+1)/2 for j>0.

T(4*j+2,4*j-2) = 4*C(j+2,4) + 6*C(j+1,2)^2 for j>1.

G.f. of row sums: (1-x-x^3)/((1-2*x)*(1-x^2)*(1+2*x^2+x^3+x^4)).

G.f. of antidiagonal sums: (1-x^2-x^3+x^4-x^6)/((1-x-x^2)*(1-x^4)*(1+3*x^4+x^8)).

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

A350112 Triangle read by rows: T(n,k) is the number of tilings of an (n+k)-board using k (1,4)-fences and n-k squares.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 2, 0, 1, 3, 6, 10, 9, 4, 0, 0, 1, 4, 10, 16, 16, 8, 0, 0, 0, 1, 5, 14, 25, 28, 16, 0, 0, 0, 0, 1, 6, 19, 38, 48, 32, 16, 8, 4, 2, 1, 1, 7, 25, 56, 80, 80, 60, 40, 25, 15, 3, 0, 1, 8, 32, 80, 136, 166, 157, 128, 95, 40, 9, 0, 0

Offset: 0

Michael A. Allen, Dec 22 2021

This is the m=5 member in the sequence of triangles A007318, A059259, A350110, A350111, A350112 which give the number of tilings of an (n+k) X 1 board using k (1,m-1)-fences and n-k unit square tiles. A (1,g)-fence is composed of two unit square tiles separated by a gap of width g.
It is also the m=5, t=2 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(5*j+r-k,k) is the coefficient of x^k in (f(j,x))^(5-r)*(f(j+1,x))^r for r=0,1,2,3,4 where f(n,x) is one form of a Fibonacci polynomial defined by f(n+1,x)=f(n,x)+x*f(n-1,x) where f(0,x)=1 and f(n<0,x)=0.
T(n+5-k,k) is the number of subsets of {1,2,...,n} of size k such that no two elements in a subset differ by 5.
Sum of (5j+r)-th antidiagonal (counting initial 1 as the 0th) is f(j)^(5-r)*f(j+1)^r where j=0,1,..., r=0,1,2,3,4, and f(n) is the Fibonacci number A000045(n+1).

			Triangle begins:
  1;
  1,   0;
  1,   0,   0;
  1,   0,   0,   0;
  1,   0,   0,   0,   0;
  1,   1,   1,   1,   1,   1;
  1,   2,   3,   4,   5,   2,   0;
  1,   3,   6,  10,   9,   4,   0,   0;
  1,   4,  10,  16,  16,   8,   0,   0,   0;
  1,   5,  14,  25,  28,  16,   0,   0,   0,   0;
  1,   6,  19,  38,  48,  32,  16,   8,   4,   2,   1;
  1,   7,  25,  56,  80,  80,  60,  40,  25,  15,   3,   0;
  1,   8,  32,  80, 136, 166, 157, 128,  95,  40,   9,   0,   0;
  1,   9,  40, 112, 217, 309, 346, 330, 223, 105,  27,   0,   0,   0;

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.
Michael A. Allen and Kenneth Edwards, On Two Families of Generalizations of Pascal's Triangle, J. Int. Seq. 25 (2022) Article 22.7.1.

Other members of the two-parameter family of triangles: A007318 (m=1,t=2), A059259 (m=2,t=2), A350110 (m=3,t=2), A350111 (m=4,t=2), A354665 (m=2,t=3), A354666 (m=2,t=4), A354667 (m=2,t=5), A354668 (m=3,t=3).

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

f[n_]:=If[n<0,0,f[n-1]+x*f[n-2]+KroneckerDelta[n,0]];
T[n_, k_]:=Module[{j=Floor[(n+k)/5], r=Mod[n+k,5]},
  Coefficient[f[j]^(5-r)*f[j+1]^r,x,k]];
Flatten@Table[T[n,k], {n, 0, 13}, {k, 0, n}]

T(n,0) = 1.
T(n,n) = delta(n mod 5,0).
T(n,1) = n-4 for n>3.
T(5*j-r,5*j-p) = 0 for j>0, p=1,2,3,4, and r=1,...,p.
T(5*(j-1)+p,5*(j-1)) = T(5*j,5*j-p) = j^p for j>0 and p=0,1,...,5.
T(5*j+1,5*j-1) = 5*j(j+1)/2 for j>0.
T(5*j+2,5*j-2) = 5*C(j+2,4) + 10*C(j+1,2)^2 for j>1.
T(n,k) = T(n-1,k) + T(n-1,k-1) for n >= 4*k+1 if k >= 0.

A354665 Triangle read by rows, T(n,k) = 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) + delta(n,0)delta(k,0) - delta(n,1)delta(k,1), T(n

Original entry on oeis.org

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

Offset: 0

Michael A. Allen, Jun 04 2022

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.

			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;
...

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.

Row sums are A011782.
Sums over k of T(n-2*k,k) are A224809.
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), A354666 (m=2,t=4), A354667 (m=2,t=5), A354668 (m=3,t=3).
Other triangles related to tiling using combs: A059259, A123521, A157897, A335964.

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

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.

A354666 Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k-1) + 2T(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

Original entry on oeis.org

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

Michael A. Allen, Jun 04 2022

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.

			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;
...

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.

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.

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

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.

A354667 Triangle read by rows: T(n,k) is the number of tilings of an (n+4*k) X 1 board using k (1,1;5)-combs and n-k squares.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 4, 0, 1, 1, 1, 6, 0, 3, 0, 1, 2, 9, 0, 9, 0, 1, 1, 3, 12, 5, 18, 0, 4, 0, 1, 4, 16, 12, 36, 0, 16, 0, 1, 1, 5, 20, 25, 60, 15, 40, 0, 5, 0, 1, 6, 25, 42, 100, 42, 100, 0, 25, 0, 1, 1, 7, 31, 66, 150, 112, 200

Offset: 0

Michael A. Allen, Jun 05 2022

This is the m=2, t=5 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-4*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 (1,5)-bonacci polynomial defined by f(n,x)=f(n-1,x)+x*f(n-5,x)+delta(n,0) where f(n<0,x)=0.
T(n+8-4*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, 6, or 8.

			Triangle begins:
  1;
  1,   0;
  1,   0,   1;
  1,   0,   2,   0;
  1,   0,   4,   0,   1;
  1,   1,   6,   0,   3,   0;
  1,   2,   9,   0,   9,   0,   1;
  1,   3,  12,   5,  18,   0,   4,   0;
  1,   4,  16,  12,  36,   0,  16,   0,   1;
  1,   5,  20,  25,  60,  15,  40,   0,   5,   0;
  1,   6,  25,  42, 100,  42, 100,   0,  25,   0,   1;
  1,   7,  31,  66, 150, 112, 200,  35,  75,   0,   6,   0;
...

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.

Row sums are A005578.
Sums over k of T(n-4*k,k) are A224811.
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), A354668 (m=3,t=3).
Other triangles related to tiling using combs: A059259, A123521, A157897, A335964.

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

T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) + 2*T(n-2,k-2) + T(n-3,k-1) - T(n-3,k-2) - 2*T(n-3,k-3) - T(n-4,k-1) + T(n-4,k-2) + T(n-4,k-3) - T(n-4,k-4) + T(n-5,k-1) - 2*T(n-5,k-3) + T(n-5,k-5) + delta(n,0)*delta(k,0) - delta(n,1)*delta(k,1) - delta(n,2)*delta(k,2) - delta(n,3)*(delta(k,1) - delta(k,3)) with T(n,k<0) = T(n

T(n,0) = 1.

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

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

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

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

G.f. of sums of T(n-4*k,k) over k: (1-x^5-x^7-x^10+x^15)/(1-x-x^5+x^6-x^7+x^8-x^9-2*x^10+x^11-x^12+2*x^15-x^16+2*x^17+x^20-x^25).

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

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.

Original entry on oeis.org

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

Michael A. Allen, Jul 30 2022

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.

			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;
...

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.

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.

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}]

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.

Showing 1-6 of 6 results.

cp's OEIS Frontend

A350111 Triangle read by rows: T(n,k) is the number of tilings of an (n+k)-board using k (1,3)-fences and n-k squares.

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A350112 Triangle read by rows: T(n,k) is the number of tilings of an (n+k)-board using k (1,4)-fences and n-k squares.

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A354665 Triangle read by rows, T(n,k) = 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) + delta(n,0)*delta(k,0) - delta(n,1)*delta(k,1), T(n

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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

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A354667 Triangle read by rows: T(n,k) is the number of tilings of an (n+4*k) X 1 board using k (1,1;5)-combs and n-k squares.

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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.

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A354665 Triangle read by rows, T(n,k) = 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) + delta(n,0)delta(k,0) - delta(n,1)delta(k,1), T(n

A354666 Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k-1) + 2T(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