A389749 Triangle read by rows: numerators of the almost-Riordan array ( (3 - 3*x)/(2*x^2 - 6*x + 3) | 3/(2*x^2 - 6*x + 3), (1 - 3*x - sqrt(5*x^2 - 6*x + 1))/(2*x) ).

1, 1, 1, 4, 2, 1, 2, 10, 5, 1, 28, 16, 58, 8, 1, 44, 76, 214, 133, 11, 1, 208, 40, 2401, 646, 235, 14, 1, 328, 568, 1029, 8920, 1393, 364, 17, 1, 1552, 896, 110677, 13400, 22549, 2536, 520, 20, 1, 272, 4240, 453527, 540124, 38485, 46933, 4156, 703, 23, 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. 19; arXiv preprint, arXiv:2406.03774 [math.CO], 2024. See p. 20.

Example

The triangle of the fractions begins as:
   1/1;
   1/1,  1/1;
   4/3,  2/1,   1/1;
   2/1, 10/3,   5/1,   1/1;
  28/9, 16/3,  58/3,   8/1,  1/1;
  44/9, 76/9, 214/3, 133/3, 11/1, 1/1;
  ...

Mathematica

T[n_, 0]:=Numerator[SeriesCoefficient[(3-3x)/(2x^2-6x+3), {x, 0, n}]]; T[n_, k_]:=Numerator[SeriesCoefficient[3/(2x^2-6x+3)((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

Crossrefs

Cf. A373744, A373746, A389706, A389708, A389710, A389739, A389750 (denominators).

Sequence in context: A280988 A175665 A200586 * A389739 A389706 A097525

Adjacent sequences: A389746 A389747 A389748 * A389750 A389751 A389752

Keyword

nonn,frac,tabl

Author

Stefano Spezia, Oct 13 2025

Status

approved