A188106 Triangle T(n,k) with the coefficient [x^k] of 1/(1-2*x-x^2+x^3)^(n-k+1) in row n, column k.

1, 1, 2, 1, 4, 5, 1, 6, 14, 11, 1, 8, 27, 42, 25, 1, 10, 44, 101, 119, 56, 1, 12, 65, 196, 342, 322, 126, 1, 14, 90, 335, 770, 1080, 847, 283, 1, 16, 119, 526, 1495, 2772, 3248, 2180, 636, 1, 18, 152, 777, 2625, 6032, 9366, 9414, 5521, 1429, 1, 20, 189, 1096, 4284, 11718, 22590, 30148, 26517, 13804, 3211

Offset: 0

Author

L. Edson Jeffery, Mar 20 2011

Keywords

• nonn
• tabl

Comments

Modified versions of the generating function for D(0)={1,2,5,11,...}=A006054(m+2), m=0,1,2,..., are related to rhombus substitution tilings (see A187068, A187069 and A187070). The columns of the triangle have generating functions 1/(1-x), 2*x/(1-x)^2, x^2*(5-x)/(1-x)^3, x^3*(11-2*x-x^2)/(1-x)^4, x^4*(25-6*x-3*x^2)/(1-x)^5, ..., for which the sum of the signed coefficients in the n-th numerator equals 2^n. The diagonals {1,2,5,...}, {1,4,14,...}, ..., are generated by successive series expansion of F(n+1,x), n=0,1,..., where F(n,x)=1/(1-2*x-x^2+x^3)^n. For example, the second diagonal is {T{1,0},T{2,1},...}={1,4,14,...}=A189426, for which successive partial sums give A189427 (excluding the zero terms). Moreover, the diagonals correspond to successive convolutions of A006054 (= the first diagonal) with itself.

Examples

			1;
1, 2;
1, 4, 5;
1, 6, 14, 11;
1, 8, 27, 42, 25;
1, 10, 44, 101, 119, 56;
1, 12, 65, 196, 342, 322, 126;
1, 14, 90, 335, 770, 1080, 847, 283;
1, 16, 119, 526, 1495 ...
		

Crossrefs

Cf. A006054, A033505, A189426, A189427.

Programs

Maple

A188106 := proc(n,k) 1/(1-2*x-x^2+x^3)^(n-k+1) ; coeftayl(%,x=0,k) ; end proc:
seq(seq(A188106(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Mar 22 2011

Formula

Sum_{k=0..n} T(n,k) = A033505(n).

T(n,0) = 1.

T(n,2) = A014106(n-1).

T(n,3) = (n-2)*(4*n^2+2*n-9)/3.

T(n,4) = (n-2)*(n-3)*(2*n+7)*(2*n-3)/6.

Extensions

a(43) and following corrected by Georg Fischer, Oct 14 2023