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A373744 Triangle read by rows: the almost-Riordan array ( 1/(1-x) | 2/((1-x)*(1+sqrt(1-4*x))), (1-2*x-sqrt(1-4*x))/(2*x) ).

%I A373744 #9 Jun 17 2024 15:50:08
%S A373744 1,1,1,1,2,1,1,4,4,1,1,9,13,6,1,1,23,41,26,8,1,1,65,131,101,43,10,1,1,
%T A373744 197,428,376,197,64,12,1,1,626,1429,1377,834,337,89,14,1,1,2056,4861,
%U A373744 5017,3382,1597,529,118,16,1,1,6918,16795,18277,13378,7105,2773,781,151,18,1
%N A373744 Triangle read by rows: the almost-Riordan array ( 1/(1-x) | 2/((1-x)*(1+sqrt(1-4*x))), (1-2*x-sqrt(1-4*x))/(2*x) ).
%H A373744 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 pp. 16-17.
%F A373744 T(n,1) = A014137(n-1).
%F A373744 T(n,n-2) = A091823(n-1) for n > 2.
%e A373744 The triangle begins as:
%e A373744   1;
%e A373744   1,   1;
%e A373744   1,   2,   1;
%e A373744   1,   4,   4,   1;
%e A373744   1,   9,  13,   6,   1;
%e A373744   1,  23,  41,  26,   8,  1;
%e A373744   1,  65, 131, 101,  43, 10,  1;
%e A373744   1, 197, 428, 376, 197, 64, 12, 1;
%e A373744   ...
%t A373744 T[n_, 0]:=1; T[n_, k_]:=SeriesCoefficient[2/((1-x)(1+Sqrt[1-4x]))((1-2x-Sqrt[1-4x])/(2x))^(k-1), {x, 0, n-1}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten
%Y A373744 Cf. A000012 (k=0 and n=k), A001453 (k=2), A004275 (subdiagonal), A014137, A091823, A143955 (k=3).
%K A373744 nonn,tabl
%O A373744 0,5
%A A373744 _Stefano Spezia_, Jun 16 2024