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%I #22 Oct 14 2023 11:29:01
%S 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,
%T 342,322,126,1,14,90,335,770,1080,847,283,1,16,119,526,1495,2772,3248,
%U 2180,636,1,18,152,777,2625,6032,9366,9414,5521,1429,1,20,189,1096,4284,11718,22590,30148,26517,13804,3211
%N 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.
%C 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.
%F Sum_{k=0..n} T(n,k) = A033505(n).
%F T(n,0) = 1.
%F T(n,2) = A014106(n-1).
%F T(n,3) = (n-2)*(4*n^2+2*n-9)/3.
%F T(n,4) = (n-2)*(n-3)*(2*n+7)*(2*n-3)/6.
%e 1;
%e 1, 2;
%e 1, 4, 5;
%e 1, 6, 14, 11;
%e 1, 8, 27, 42, 25;
%e 1, 10, 44, 101, 119, 56;
%e 1, 12, 65, 196, 342, 322, 126;
%e 1, 14, 90, 335, 770, 1080, 847, 283;
%e 1, 16, 119, 526, 1495 ...
%p A188106 := proc(n,k) 1/(1-2*x-x^2+x^3)^(n-k+1) ; coeftayl(%,x=0,k) ; end proc:
%p seq(seq(A188106(n,k),k=0..n),n=0..10) ; # _R. J. Mathar_, Mar 22 2011
%Y Cf. A006054, A033505, A189426, A189427.
%K nonn,tabl
%O 0,3
%A _L. Edson Jeffery_, Mar 20 2011
%E a(43) and following corrected by _Georg Fischer_, Oct 14 2023
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