A389707 Triangle read by rows: denominators of the almost-Riordan array ( 3*(7 - 4*x + sqrt(1 - 8*x))/(24 - 48*x + 16*x^2 + (3*x - 3)*(1 - 4*x - sqrt(1 - 8*x))) | 24/(24 - 48*x + 16*x^2 + (3*x - 3)*(1 - 4*x - sqrt(1 - 8*x))), (1 - 4*x - sqrt(1 - 8*x))/(8*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, 1, 3, 1, 1, 27, 27, 9, 9, 3, 3, 1, 1, 81, 27, 27, 9, 9, 3, 3, 1, 1, 9, 81, 9, 27, 1, 9, 1, 3, 1, 1, 243, 81, 81, 27, 27, 9, 9, 3, 3, 1, 1, 243, 243, 81, 81, 27, 27, 9, 9, 3, 3, 1, 1, 729, 81, 243, 27, 81, 3, 27, 3, 9, 1, 3, 1, 1

Offset

0,4

Comments

Differs from A389740 from the 21st row onward.

Links

Stefano Spezia, Table of n, a(n) for n = 0..5150, (rows 0..100 of the triangle)

Tian-Xiao He and Roksana Słowik, Total Positivity of Almost-Riordan Arrays, Graphs and Combinatorics 41, 115 (2025), see pp. 15-16; arXiv preprint, arXiv:2406.03774 [math.CO], 2024. See p. 17.

Example

The triangle of the fractions begins as:
   1/1;
   1/1,   1/1;
   4/3,   2/1,   1/1;
   2/1,  13/2,   6/1,   1/1;
  31/9,  37/3,  97/3,  10/1,  1/1;
  68/9, 433/9, 545/3, 229/3, 14/1, 1/1;
  ...

Mathematica

T[n_, 0]:=Denominator[SeriesCoefficient[3(7-4x+Sqrt[1-8x])/(24-48x+16x^2+(3x-3)(1-4x-Sqrt[1-8x])), {x, 0, n}]]; T[n_, k_]:=Denominator[SeriesCoefficient[24/(24-48x+16x^2+(3x-3)(1-4x-Sqrt[1-8x]))*((1-4x-Sqrt[1-8x])/(8x))^(k-1), {x, 0, n-1}]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten

Crossrefs

Cf. A373744, A373746, A389706 (numerators), A389709, A389711, A389740.

Sequence in context: A226203 A327791 A051997 * A389740 A389750 A155744

Adjacent sequences: A389704 A389705 A389706 * A389708 A389709 A389710

Keyword

nonn,frac,tabl

Author

Stefano Spezia, Oct 12 2025

Status

approved