A367780 a(n) is the sum of the squares of the area under Dyck paths of length 2*n.

0, 1, 20, 189, 1356, 8426, 47944, 257085, 1321036, 6574190, 31911320, 151841906, 710828600, 3282862644, 14988894992, 67769474077, 303823057164, 1352059744070, 5977826290936, 26277396651558, 114916296684008, 500229317398156, 2168403190878960, 9364025672275634

Offset: 0

Author

AJ Bu, Nov 29 2023

Keywords

• nonn

Links

• AJ Bu, Explicit Generating Functions for the Sum of the Areas Under Dyck and Motzkin Paths (and for Their Powers), arXiv:2310.17026 [math.CO], 2023.

Crossrefs

Cf. A000108, A008549.

Programs

Maple

G:= ((-1 + sqrt(-4*x^2 + 1))*(40*x^4 + 14*sqrt(-4*x^2 + 1)*x^2 - 14*x^2 - sqrt(-4*x^2 + 1) + 1))/( 4*(4*x^2 - 1)^3*x^2):  Gser:=series(G, x=0, 41): seq(coeff(Gser, x, 2*n), n=0..19);

Mathematica

G[x_] := ((-1 + Sqrt[-4*x^2 + 1]) * (40*x^4 + 14*Sqrt[-4*x^2 + 1]*x^2 - 14*x^2 - Sqrt[-4*x^2 + 1] + 1)) /  (4*(4*x^2 - 1)^3*x^2); Gser = Series[G[x], {x, 0, 46}]; Table[Coefficient[Gser, x, 2*n], {n, 0, 23}] (* James C. McMahon, Dec 10 2023 *)

Formula

G.f.: ((-1 + sqrt(-4*x^2 + 1))*(40*x^4 + 14*sqrt(-4*x^2 + 1)*x^2 - 14*x^2 - sqrt(-4*x^2 + 1) + 1))/( 4*(4*x^2 - 1)^3*x^2).

D-finite with recurrence -(n+1)*(133*n-262)*a(n) +4*(564*n^2-1229*n+262)*a(n-1) +4*(-2916*n^2+7294*n-2765)*a(n-2) +16*(596*n-553)*(2*n-3)*a(n-3)=0. - R. J. Mathar, Jan 11 2024