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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.

Second of a series of sequences of partial sums of (nonzero) diagonals of triangle A188106 whose diagonals correspond to successive convolutions of A006054 with itself, where the first such sequence of partial sums is given by A077850. For n=1,2,..., this series of sequences is generated by successive series expansion of 1/((1-x)*(1-2*x-x^2+x^3)^n), for which A077850 corresponds to n=1 and A189427 corresponds to n=2.

a(n)=Sum_{k=0..n} A189426(k), where A189426={0,0,1,4,14,42,119,322,...} is the convolution of A006054={0,0,1,2,5,11,25,56,126,...} with itself. Also, a(n+2)=Sum_{k=0..n} A188106{n+k+1,k}, n=0,1,2,....