A389709 Triangle read by rows: denominators 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 ).

1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 8, 1, 1, 1, 1, 8, 8, 1, 4, 1, 1, 16, 4, 16, 2, 2, 1, 1, 16, 64, 16, 16, 2, 4, 1, 1, 128, 8, 8, 2, 4, 1, 1, 1, 1, 128, 128, 8, 16, 4, 4, 1, 4, 1, 1, 256, 64, 128, 8, 8, 2, 16, 2, 2, 1, 1, 256, 512, 128, 128, 8, 64, 16, 16, 2, 4, 1, 1, 1024, 128, 128, 32, 256, 16, 16, 4, 8, 1, 1, 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 pp. 16-17; arXiv preprint, arXiv:2406.03774 [math.CO], 2024. See pp. 17-18.

Example

The triangle of the fractions begins as:
   1/1;
   1/1,   1/1;
   3/2,   2/1,  1/1;
   5/2,  15/4,  3/1,  1/1;
  35/8,   7/1,  7/1,  4/1, 1/1;
  63/8, 105/8, 15/1, 45/4, 5/1, 1/1;
  ...

Mathematica

T[n_, 0]:=Denominator[SeriesCoefficient[(1-x+Sqrt[1-2x])/(1-2x+(1-x)Sqrt[1-2x]), {x, 0, n}]]; T[n_, k_]:=Denominator[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, 12}, {k, 0, n}]//Flatten

Crossrefs

Cf. A373744, A373746, A389707, A389708 (numerators), A389711, A389740.

Sequence in context: A166288 A327001 A249138 * A376555 A056648 A056061

Adjacent sequences: A389706 A389707 A389708 * A389710 A389711 A389712

Keyword

nonn,frac,tabl

Author

Stefano Spezia, Oct 12 2025

Status

approved