A389750 Triangle read by rows: denominators 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, 3, 1, 1, 1, 3, 1, 1, 9, 3, 3, 1, 1, 9, 9, 3, 3, 1, 1, 27, 3, 9, 3, 3, 1, 1, 27, 27, 1, 9, 3, 3, 1, 1, 81, 27, 27, 3, 9, 3, 3, 1, 1, 9, 81, 27, 27, 3, 9, 3, 3, 1, 1, 243, 81, 81, 27, 27, 1, 9, 3, 3, 1, 1, 243, 243, 81, 81, 27, 27, 3, 9, 3, 3, 1, 1, 729, 81, 243, 81, 81, 27, 27, 3, 9, 3, 3, 1, 1

Offset

0,4

Links

Table of n, a(n) for n=0..90.

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]:=Denominator[SeriesCoefficient[(3-3x)/(2x^2-6x+3), {x, 0, n}]]; T[n_, k_]:=Denominator[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, A389707, A389709, A389711, A389740, A389749 (numerators).

Sequence in context: A051997 A389707 A389740 * A155744 A086869 A348953

Adjacent sequences: A389747 A389748 A389749 * A389751 A389752 A389753

Keyword

nonn,frac,tabl

Author

Stefano Spezia, Oct 13 2025

Status

approved