A389708 - OEIS

%I #12 Oct 13 2025 03:09:08

%S 1,1,1,3,2,1,5,15,3,1,35,7,7,4,1,63,105,15,45,5,1,231,99,495,55,33,6,

%T 1,429,3003,1001,1001,91,91,7,1,6435,715,1001,273,455,70,30,8,1,12155,

%U 21879,1989,4641,1071,765,102,153,9,1,46189,20995,62985,4845,4845,969,4845,285,95,10,1

%N Triangle read by rows: numerators of the almost-Riordan array ( (1 - x + sqrt(1 - 2*x))/(1 - 2*x + (1 - x)*sqrt(1 - 2*x)) | 2/(1 - 2*x + (1 - x)*sqrt(1 - 2*x)), 2*(1 - x - sqrt(1 - 2*x))/x ).

%H Tian-Xiao He and Roksana Słowik, <a href="https://doi.org/10.1007/s00373-025-02979-6">Total Positivity of Almost-Riordan Arrays</a>, Graphs and Combinatorics 41, 115 (2025), see pp. 16-17; <a href="https://arxiv.org/abs/2406.03774">arXiv preprint</a>, arXiv:2406.03774 [math.CO], 2024. See pp. 17-18.

%e The triangle of the fractions begins as:

%e    1/1;

%e    1/1,   1/1;

%e    3/2,   2/1,  1/1;

%e    5/2,  15/4,  3/1,  1/1;

%e   35/8,   7/1,  7/1,  4/1, 1/1;

%e   63/8, 105/8, 15/1, 45/4, 5/1, 1/1;

%e   ...

%t T[n_, 0]:=Numerator[SeriesCoefficient[(1-x+Sqrt[1-2x])/(1-2x+(1-x)Sqrt[1-2x]),{x,0,n}]]; T[n_, k_]:=Numerator[SeriesCoefficient[2/(1-2x+(1-x)Sqrt[1-2x])*(2(1-x-Sqrt[1-2x])/x)^(k-1), {x, 0, n-1}]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten

%Y Cf. A373744, A373746, A389706, A389709 (denominators), A389710, A389739.

%K nonn,frac,tabl

%O 0,4

%A _Stefano Spezia_, Oct 12 2025