This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A361038 #14 Mar 07 2023 11:54:40 %S A361038 280,210,420,1176,3960,15015,61880,271320,1248072,5965050,29414700, %T A361038 148874400,770263200,4061212722,21765976680,118336861720,651555929640, %U A361038 3627981880950,20405547069180,115815267149400,662742214356600 %N A361038 a(n) = 1680 * (3*n)!/((2*n)!*(n+3)!). %C A361038 Compare with the super ballot numbers A007272(n) = 60*(2*n)!/(n!*(n+3)!). %F A361038 a(n) = 280*binomial(3*n,n) - 228*binomial(3*n,n+1) + 54*binomial(3*n,n+2) - 5*binomial(3*n,n+3). Thus a(n) is an integer. %F A361038 P-recursive: 2*(n + 3)*(2*n - 1) = 3*(3*n - 1)*(3*n - 2)*a(n-1) with a(0) = 280. %F A361038 a(n) ~ (27/4)^n * 840*sqrt(3/Pi)/n^(7/2). %F A361038 The o.g.f. satisfies the differential equation %F A361038 x^2*(27*x - 4)*A''(x) + 2*x*(27*x - 7)*A'(x) + (6*x + 6)*A(x) - 1680 = 0, with A(0) = 280 and A'(0) = 210. %p A361038 seq( 1680 * (3*n)!/((2*n)!*(n+3)!), n = 0..20); %Y A361038 Cf. A000139, A001764, A007226, A007272, A361037, A361039, A361040. %K A361038 nonn,easy %O A361038 0,1 %A A361038 _Peter Bala_, Mar 04 2023