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 A361039 #14 Mar 07 2023 11:54:37 %S A361039 2310,1386,2310,5544,16335,55055,204204,813960,3432198,15142050, %T A361039 69334650,327523680,1588667850,7883530578,39904290580,205532444040, %U A361039 1075067283906,5701114384350,30608320603770,166169731127400,911270544740325 %N A361039 a(n) = 55440 * (3*n)!/((2*n)!*(n+4)!). %C A361039 Compare with the super ballot numbers A348893(n) = 840*(2*n)!/(n!*(n+4)!). %F A361039 a(n) = 2310*binomial(3*n,n) - 2057*binomial(3*n,n+1) + 627*binomial(3*n,n+2) - 102*binomial(3*n,n+3) + 7*binomial(3*n, n+4). Thus a(n) is an integer. %F A361039 P-recursive: 2*(n + 4)*(2*n - 1) = 3*(3*n - 1)*(3*n - 2)*a(n-1) with a(0) = 2310. %F A361039 a(n) ~ (27/4)^n * 27720*sqrt(3/Pi)/n^(9/2). %F A361039 The o.g.f. satisfies the differential equation %F A361039 x^2*(27*x - 4)*A''(x) + 2*x*(27*x - 9)*A'(x) + (6*x + 8)*A(x) - 18480 = 0, with A(0) = 2310 and A'(0) = 1386. %p A361039 seq( 55440 * (3*n)!/((2*n)!*(n+4)!), n = 0..20); %Y A361039 Cf. A000139, A001764, A007226, A007272, A361037, A361038, A361041. %K A361039 nonn,easy %O A361039 0,1 %A A361039 _Peter Bala_, Mar 04 2023