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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<k,k) = T(n,k<0) = 0.
6
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
0,9
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.
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.
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.
EXAMPLE
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;
...
MATHEMATICA
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
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.
Sequence in context: A102210 A124220 A221755 * A110298 A362379 A144740
KEYWORD
easy,nonn,tabl
AUTHOR
Michael A. Allen, Jun 04 2022
STATUS
approved