login
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) ).
0

%I #9 Jun 17 2024 15:50:08

%S 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 197,428,376,197,64,12,1,1,626,1429,1377,834,337,89,14,1,1,2056,4861,

%U 5017,3382,1597,529,118,16,1,1,6918,16795,18277,13378,7105,2773,781,151,18,1

%N 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 Tian-Xiao He and Roksana SÅ‚owik, <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 T(n,1) = A014137(n-1).

%F T(n,n-2) = A091823(n-1) for n > 2.

%e The triangle begins as:

%e 1;

%e 1, 1;

%e 1, 2, 1;

%e 1, 4, 4, 1;

%e 1, 9, 13, 6, 1;

%e 1, 23, 41, 26, 8, 1;

%e 1, 65, 131, 101, 43, 10, 1;

%e 1, 197, 428, 376, 197, 64, 12, 1;

%e ...

%t 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 Cf. A000012 (k=0 and n=k), A001453 (k=2), A004275 (subdiagonal), A014137, A091823, A143955 (k=3).

%K nonn,tabl

%O 0,5

%A _Stefano Spezia_, Jun 16 2024