cp's OEIS Frontend

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

%I A367780 #23 Jan 11 2024 11:06:15
%S A367780 0,1,20,189,1356,8426,47944,257085,1321036,6574190,31911320,151841906,
%T A367780 710828600,3282862644,14988894992,67769474077,303823057164,
%U A367780 1352059744070,5977826290936,26277396651558,114916296684008,500229317398156,2168403190878960,9364025672275634
%N A367780 a(n) is the sum of the squares of the area under Dyck paths of length 2*n.
%H A367780 AJ Bu, <a href="https://arxiv.org/abs/2310.17026">Explicit Generating Functions for the Sum of the Areas Under Dyck and Motzkin Paths (and for Their Powers)</a>, arXiv:2310.17026 [math.CO], 2023.
%F A367780 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).
%F A367780 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
%p A367780 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);
%t A367780 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 *)
%Y A367780 Cf. A000108, A008549.
%K A367780 nonn
%O A367780 0,3
%A A367780 _AJ Bu_, Nov 29 2023