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.
FORMULA
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
KEYWORD
nonn,tabl
AUTHOR
L. Edson Jeffery, Mar 20 2011
EXTENSIONS
a(43) and following corrected by Georg Fischer, Oct 14 2023
STATUS
approved