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A201780 Riordan array ((1-x)^2/(1-2x), x/(1-2x)).

%I A201780 #25 Nov 04 2018 01:24:27
%S A201780 1,0,1,1,2,1,2,5,4,1,4,12,13,6,1,8,28,38,25,8,1,16,64,104,88,41,10,1,
%T A201780 32,144,272,280,170,61,12,1,64,320,688,832,620,292,85,14,1,128,704,
%U A201780 1696,2352,2072,1204,462,113,16,1
%N A201780 Riordan array ((1-x)^2/(1-2x), x/(1-2x)).
%C A201780 Diagonals ascending: 1, 0, 1, 1, 2, 2, 4, 5, 1, 8, 12, 4, ... (see A201509).
%H A201780 Benjamin Braun, W. K. Hough, <a href="https://arxiv.org/abs/1606.01204">Matching and Independence Complexes Related to Small Grids</a>, arXiv preprint arXiv:1606.01204 [math.CO], 2016.
%H A201780 Wesley K. Hough, <a href="https://dx.doi.org/10.13023/ETD.2017.119">On Independence, Matching, and Homomorphism Complexes</a>, (2017), Theses and Dissertations--Mathematics, 42.
%F A201780 T(n,k) = 2*T(n-1,k) + T(n-1,k-1) with T(0,0) = 0, T(1,0) = 0, T(2,0) = 0 and T(n,k)= 0 if k < 0 or if n < k.
%F A201780 Sum_{k=0..n} T(n,k)*x^k = A154955(n+1), A034008(n), A052156(n), A055841(n), A055842(n), A055846(n), A055270(n), A055847(n), A055995(n), A055996(n), A056002(n), A056116(n) for x = -1,0,1,2,3,4,5,6,7,8,9,10 respectively.
%F A201780 G.f.: (1-x)^2/(1-(y+2)*x).
%e A201780 Triangle begins:
%e A201780   1;
%e A201780   0,  1;
%e A201780   1,  2,  1;
%e A201780   2,  5,  4,  1;
%e A201780   4, 12, 13,  6,  1;
%e A201780   8, 28, 38, 25,  8,  1;
%t A201780 CoefficientList[#, y]& /@ CoefficientList[(1-x)^2/(1-(y+2)*x) + O[x]^10, x] // Flatten (* _Jean-François Alcover_, Nov 03 2018 *)
%Y A201780 Diagonals: A000012, A005843, A001844, A035597,
%Y A201780 Columns: A034008, A045623, A084851, A055585,
%Y A201780 Row sums: A052156
%K A201780 nonn,tabl
%O A201780 0,5
%A A201780 _Philippe Deléham_, Dec 05 2011