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A373746 Triangle read by rows: the almost-Riordan array ( 1/(1-x) | 2/((1-x)*(1+x+sqrt(5*x^2-6*x+1))), (1-3*x-sqrt(5*x^2-6*x+1))/(2x) ).

%I A373746 #5 Jun 17 2024 15:50:16
%S A373746 1,1,1,1,2,1,1,4,5,1,1,10,20,8,1,1,31,78,45,11,1,1,110,310,224,79,14,
%T A373746 1,1,421,1264,1061,475,122,17,1,1,1686,5274,4922,2608,858,174,20,1,1,
%U A373746 6961,22430,22648,13604,5356,1400,235,23,1,1,29392,96899,103978,68816,31072,9791,2128,305,26,1
%N A373746 Triangle read by rows: the almost-Riordan array ( 1/(1-x) | 2/((1-x)*(1+x+sqrt(5*x^2-6*x+1))), (1-3*x-sqrt(5*x^2-6*x+1))/(2x) ).
%C A373746 In He and Słowik, there is a typing error since T(5,1) is equal to 31 and not to 421.
%H A373746 Tian-Xiao He and Roksana Słowik, <a href="https://arxiv.org/abs/2406.03774">Total Positivity of Almost-Riordan Arrays</a>, arXiv:2406.03774 [math.CO], 2024. See p. 19.
%F A373746 T(n,n-1) = A016789(n-2).
%e A373746 The triangle begins as:
%e A373746   1;
%e A373746   1,   1;
%e A373746   1,   2,   1;
%e A373746   1,   4,   5,   1;
%e A373746   1,  10,  20,   8,  1;
%e A373746   1,  31,  78,  45, 11,  1;
%e A373746   1, 110, 310, 224, 79, 14, 1;
%e A373746   ...
%t A373746 T[n_, 0]:=1; T[n_, k_]:=SeriesCoefficient[2/((1-x)(1+x+Sqrt[5x^2-6x+1]))((1-3x-Sqrt[5x^2-6x+1])/(2x))^(k-1), {x, 0, n-1}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten
%Y A373746 Cf. A000012 (k=0 and n=k), A016789, A138415 (k=1).
%K A373746 nonn,tabl
%O A373746 0,5
%A A373746 _Stefano Spezia_, Jun 16 2024