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A188107 Triangle T(n,k) with the coefficient [x^k] of 1/(1 - x - 2*x^2 + x^3)^(n-k+1) in row n, column k.

%I A188107 #19 Feb 01 2019 02:34:35
%S A188107 1,1,1,1,2,3,1,3,7,4,1,4,12,14,9,1,5,18,31,35,14,1,6,25,56,87,70,28,1,
%T A188107 7,33,90,175,207,154,47,1,8,42,134,310,476,504,306,89,1,9,52,189,504,
%U A188107 941,1274,1137,633,155,1,10,63,256,770,1680,2745,3188,2571
%N A188107 Triangle T(n,k) with the coefficient [x^k] of 1/(1 - x - 2*x^2 + x^3)^(n-k+1) in row n, column k.
%C A188107 Modified versions of the generating function for the diagonal, A006053, are related to rhombus substitution tilings (see A187065, A187066 and A187067).
%H A188107 Nathaniel Johnston, <a href="/A188107/b188107.txt">Rows n = 0..100, flattened</a>
%F A188107 Sum_{k=0..n} T(n,k) = A001654(n+1).
%F A188107 T(n,k) = T(n-1,k) + T(n-1,k-1) + 2*T(n-2,k-2) - T(n-3,k-3). - _Philippe Deléham_, Feb 24 2012
%e A188107 The triangle starts in row n=0 as
%e A188107   1;
%e A188107   1,   1;
%e A188107   1,   2,   3;
%e A188107   1,   3,   7,   4;
%e A188107   1,   4,  12,  14,   9;
%e A188107   1,   5,  18,  31,  35,  14;
%e A188107   1,   6,  25,  56,  87,  70,  28;
%e A188107   1,   7,  33,  90, 175, 207, 154,  47;
%e A188107   1,   8,  42, 134, 310, 476, 504, 306,  89;
%p A188107 A188107 := proc(n,k) 1/(1-x-2*x^2+x^3)^(n-k+1) ; coeftayl(%,x=0,k) ; end proc:
%p A188107 seq(seq(A188107(n,k),k=0..n),n=0..10) ; # _R. J. Mathar_, Mar 22 2011
%Y A188107 Cf. A001654, A187065, A187066, A187067.
%Y A188107 Cf. Columns: A000012, A000027, A055998.
%K A188107 nonn,easy,tabl
%O A188107 0,5
%A A188107 _L. Edson Jeffery_, Mar 20 2011