A373744 Triangle read by rows: the almost-Riordan array ( 1/(1-x) | 2/((1-x)*(1+sqrt(1-4*x))), (1-2*x-sqrt(1-4*x))/(2*x) ).

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 9, 13, 6, 1, 1, 23, 41, 26, 8, 1, 1, 65, 131, 101, 43, 10, 1, 1, 197, 428, 376, 197, 64, 12, 1, 1, 626, 1429, 1377, 834, 337, 89, 14, 1, 1, 2056, 4861, 5017, 3382, 1597, 529, 118, 16, 1, 1, 6918, 16795, 18277, 13378, 7105, 2773, 781, 151, 18, 1

Offset: 0

Author

Stefano Spezia, Jun 16 2024

Keywords

• nonn
• tabl

Examples

			The triangle begins as:
  1;
  1,   1;
  1,   2,   1;
  1,   4,   4,   1;
  1,   9,  13,   6,   1;
  1,  23,  41,  26,   8,  1;
  1,  65, 131, 101,  43, 10,  1;
  1, 197, 428, 376, 197, 64, 12, 1;
  ...
		

Links

• Tian-Xiao He and Roksana Słowik, Total Positivity of Almost-Riordan Arrays, arXiv:2406.03774 [math.CO], 2024. See pp. 16-17.

Crossrefs

Cf. A000012 (k=0 and n=k), A001453 (k=2), A004275 (subdiagonal), A014137, A091823, A143955 (k=3).

Programs

Mathematica

T[n_, 0]:=1; T[n_, k_]:=SeriesCoefficient[2/((1-x)(1+Sqrt[1-4x]))((1-2x-Sqrt[1-4x])/(2x))^(k-1), {x, 0, n-1}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten

Formula

T(n,1) = A014137(n-1).

T(n,n-2) = A091823(n-1) for n > 2.