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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<k,k) = T(n,k<0) = 0.
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%I #12 Dec 27 2022 03:26:32

%S 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,

%T 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,

%U 81,100,20,25,0,1,1,8,34,84,158,168

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

%C 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.

%C 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.

%C 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.

%H Michael A. Allen, <a href="https://arxiv.org/abs/2209.01377">On a Two-Parameter Family of Generalizations of Pascal's Triangle</a>, arXiv:2209.01377 [math.CO], 2022.

%H Michael A. Allen, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Allen2/allen8.html">On A Two-Parameter Family of Generalizations of Pascal's Triangle</a>, J. Int. Seq. 25 (2022) Article 22.9.8.

%F T(n,0) = 1.

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

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

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

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

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

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

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

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

%F 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).

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

%e Triangle begins:

%e 1;

%e 1, 0;

%e 1, 0, 1;

%e 1, 0, 2, 0;

%e 1, 1, 4, 0, 1;

%e 1, 2, 6, 0, 3, 0;

%e 1, 3, 9, 4, 9, 0, 1;

%e 1, 4, 12, 10, 18, 0, 4, 0;

%e 1, 5, 16, 21, 36, 10, 16, 0, 1;

%e 1, 6, 21, 36, 60, 30, 40, 0, 5, 0;

%e 1, 7, 27, 57, 100, 81, 100, 20, 25, 0, 1;

%e 1, 8, 34, 84, 158, 168, 200, 70, 75, 0, 6, 0;

%e 1, 9, 42, 118, 243, 322, 400, 231, 225, 35, 36, 0, 1;

%e ...

%t T[n_,k_]:=If[k<0 || n<k, 0, 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] + KroneckerDelta[n, k, 0] - KroneckerDelta[n,2]*(KroneckerDelta[k,1]+KroneckerDelta[k,2])]; Table[T[n,k], {n, 0, 11}, {k, 0, n}]//TableForm

%Y Row sums are A099163.

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

%Y 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).

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

%K easy,nonn,tabl

%O 0,9

%A _Michael A. Allen_, Jun 04 2022