A373746 Triangle read by rows: the almost-Riordan array ( 1/(1-x) | 2/((1-x)*(1+x+sqrt(5*x^2-6*x+1))), (1-3*x-sqrt(5*x^2-6*x+1))/(2x) ).

1, 1, 1, 1, 2, 1, 1, 4, 5, 1, 1, 10, 20, 8, 1, 1, 31, 78, 45, 11, 1, 1, 110, 310, 224, 79, 14, 1, 1, 421, 1264, 1061, 475, 122, 17, 1, 1, 1686, 5274, 4922, 2608, 858, 174, 20, 1, 1, 6961, 22430, 22648, 13604, 5356, 1400, 235, 23, 1, 1, 29392, 96899, 103978, 68816, 31072, 9791, 2128, 305, 26, 1

Offset: 0

Author

Stefano Spezia, Jun 16 2024

Keywords

• nonn
• tabl

Comments

In He and Słowik, there is a typing error since T(5,1) is equal to 31 and not to 421.

Examples

			The triangle begins as:
  1;
  1,   1;
  1,   2,   1;
  1,   4,   5,   1;
  1,  10,  20,   8,  1;
  1,  31,  78,  45, 11,  1;
  1, 110, 310, 224, 79, 14, 1;
  ...
		

Links

• Tian-Xiao He and Roksana Słowik, Total Positivity of Almost-Riordan Arrays, arXiv:2406.03774 [math.CO], 2024. See p. 19.

Crossrefs

Cf. A000012 (k=0 and n=k), A016789, A138415 (k=1).

Programs

Mathematica

T[n_, 0]:=1; T[n_, k_]:=SeriesCoefficient[2/((1-x)(1+x+Sqrt[5x^2-6x+1]))((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

Formula

T(n,n-1) = A016789(n-2).