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<k,k) = T(n,k<0) = 0.

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

0,9

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

Table of n, a(n) for n=0..70.

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

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.

Sequence in context: A354773 A368724 A304784 * A131644 A115346 A140531

Adjacent sequences: A354662 A354663 A354664 * A354666 A354667 A354668

Keyword

easy,nonn,tabl

Author

Michael A. Allen, Jun 04 2022

Status

approved