A389708 Triangle read by rows: numerators 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, 3, 2, 1, 5, 15, 3, 1, 35, 7, 7, 4, 1, 63, 105, 15, 45, 5, 1, 231, 99, 495, 55, 33, 6, 1, 429, 3003, 1001, 1001, 91, 91, 7, 1, 6435, 715, 1001, 273, 455, 70, 30, 8, 1, 12155, 21879, 1989, 4641, 1071, 765, 102, 153, 9, 1, 46189, 20995, 62985, 4845, 4845, 969, 4845, 285, 95, 10, 1

Offset

0,4

Links

Table of n, a(n) for n=0..65.

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]:=Numerator[SeriesCoefficient[(1-x+Sqrt[1-2x])/(1-2x+(1-x)Sqrt[1-2x]), {x, 0, n}]]; T[n_, k_]:=Numerator[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, 10}, {k, 0, n}]//Flatten

Crossrefs

Cf. A373744, A373746, A389706, A389709 (denominators), A389710, A389739.

Sequence in context: A108123 A105954 A144252 * A396676 A248033 A318254

Adjacent sequences: A389705 A389706 A389707 * A389709 A389710 A389711

Keyword

nonn,frac,tabl

Author

Stefano Spezia, Oct 12 2025

Status

approved