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.

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

0,9

Comments

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.

Links

Table of n, a(n) for n=0..72.

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,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<k,k) = 0.

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.

Example

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;
...

Mathematica

T[n_, k_]:=If[k<0 || n<k, 0, 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] + KroneckerDelta[n, k, 0] - KroneckerDelta[n, k, 1] - KroneckerDelta[n, k, 2] - KroneckerDelta[n, 3]*KroneckerDelta[k, 1] + KroneckerDelta[n, k, 3]]; Flatten@Table[T[n, k], {n, 0, 11}, {k, 0, n}]
f[n_]:=If[n<0, 0, f[n-1]+x*f[n-5]+KroneckerDelta[n, 0]]; T[n_, k_]:=Module[{j=Floor[(n+4*k)/2], r=Mod[n+4*k, 2]}, Coefficient[f[j]^(2-r)*f[j+1]^r, x, k]]; Flatten@Table[T[n, k], {n, 0, 11}, {k, 0, n}]

Crossrefs

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.

Sequence in context: A073430 A371093 A053389 * A202328 A136688 A131321

Adjacent sequences: A354664 A354665 A354666 * A354668 A354669 A354670

Keyword

easy,nonn,tabl

Author

Michael A. Allen, Jun 05 2022

Status

approved