A373744 - OEIS

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

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OFFSET

0,5

LINKS

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

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

FORMULA

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

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

EXAMPLE

The triangle begins as:

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;

...

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

CROSSREFS

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

Sequence in context: A155971 A176480 A154218 * A326326 A307139 A078121

Adjacent sequences: A373741 A373742 A373743 * A373745 A373746 A373747

KEYWORD

nonn,tabl

AUTHOR

Stefano Spezia, Jun 16 2024

STATUS

approved