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

0,3

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.

Links

Table of n, a(n) for n=0..65.

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.

Example

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 ...

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

Crossrefs

Cf. A006054, A033505, A189426, A189427.

Sequence in context: A161135 A237274 A038730 * A050166 A124959 A081281

Adjacent sequences: A188103 A188104 A188105 * A188107 A188108 A188109

Keyword

nonn,tabl

Author

L. Edson Jeffery, Mar 20 2011

Extensions

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

Status

approved