A389710 Triangle read by rows: numerators of the almost-Riordan array ( (3*x - 2 - 2*sqrt(1 - x))/(-x^2 + 5*x - 2 + 2*(x - 1)*sqrt(1 - x)) | 4/(2*(1 - x)*sqrt(1 - x) + x^2 - 5*x + 2), (8 - 4*x - 8*sqrt(1 - x))/x ).

1, 1, 1, 3, 2, 1, 5, 57, 5, 1, 137, 199, 39, 3, 1, 473, 1383, 283, 103, 7, 1, 3275, 9599, 4001, 393, 33, 4, 1, 22699, 133193, 13993, 5705, 533, 165, 9, 1, 314785, 461971, 97435, 20231, 4001, 707, 101, 5, 1, 1091541, 3204459, 42317, 70949, 7235, 1379, 919, 243, 11, 1

Offset

0,4

Links

Table of n, a(n) for n=0..54.

Tian-Xiao He and Roksana Słowik, Total Positivity of Almost-Riordan Arrays, Graphs and Combinatorics 41, 115 (2025), see p. 17; arXiv preprint, arXiv:2406.03774 [math.CO], 2024. See p. 18.

Example

The triangle of the fractions begins as:
     1/1;
     1/1,      1/1;
     3/2,      2/1,    1/1;
     5/2,    57/16,    5/2,    1/1;
  137/32,   199/32,   39/8,    3/1, 1/1;
  473/64, 1383/128, 283/32, 103/16, 7/2, 1/1;
  ...

Mathematica

T[n_, 0]:=Numerator[SeriesCoefficient[(3x-2-2Sqrt[1-x])/(-x^2+5x-2+2(x-1)Sqrt[1-x]), {x, 0, n}]]; T[n_, k_]:=Numerator[SeriesCoefficient[4/(2(1-x)Sqrt[1-x]+x^2-5x+2)*((8-4x-8Sqrt[1-x])/x)^(k-1), {x, 0, n-1}]]; Table[T[n, k], {n, 0, 9}, {k, 0, n}]//Flatten

Crossrefs

Cf. A373744, A373746, A389706, A389708, A389711 (denominators), A389739.

Sequence in context: A318254 A002130 A382347 * A089145 A324644 A364256

Adjacent sequences: A389707 A389708 A389709 * A389711 A389712 A389713

Keyword

nonn,frac,tabl

Author

Stefano Spezia, Oct 12 2025

Status

approved