A382668 a(n) = C(n+1) - C(n-1) - 2*C(n-2) where C(n) = A000108(n) are the Catalan numbers.

2, 10, 33, 108, 359, 1214, 4169, 14508, 51064, 181492, 650522, 2348856, 8535921, 31197430, 114601065, 422891340, 1566903060, 5827192140, 21743726430, 81383916840, 305465105790, 1149489049644, 4335921660522, 16391329697528, 62091796219904, 235656705875304

Offset: 2

Author

F. Chapoton, Apr 02 2025

Keywords

• nonn
• easy

Crossrefs

Cf. A000108, A026012, A280891.

Programs

Maple

gf := ((2*x^3 + x^2 - 1)*sqrt(1 - 4*x) - 4*x^3 - 3*x^2 - 2*x + 1)/(2*x^2):
ser := series(gf, x, 30): seq(coeff(ser, x, n), n = 2..27);  # Peter Luschny, Apr 03 2025

Mathematica

a[n_]:=CatalanNumber[n+1]-CatalanNumber[n-1]-2CatalanNumber[n-2];Array[a,26,2] (* James C. McMahon, Apr 05 2025 *)

SageMath

C = catalan_number
[C(n + 1) - C(n - 1) - 2 * C(n - 2) for n in range(2, 28)]

Formula

a(n) = [x^n] ((2*x^3 + x^2 - 1)*sqrt(1 - 4*x) - 4*x^3 - 3*x^2 - 2*x + 1)/(2*x^2). - Peter Luschny, Apr 03 2025