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 A361035 #9 Mar 15 2023 12:40:52 %S A361035 46200,17325,116424,2134440,67953600,3086579925,179961581800, %T A361035 12633303042360,1023952465972800,93080123469333000, %U A361035 9292590788015304000,1003030870975774344000,115656146295979953692160,14112534648127632044761125,1808633485822731984665865000 %N A361035 a(n) = 9979200 * (4*n)!/(n!*(n+3)!^3). %C A361035 Row 2 of A361032. %C A361035 The central binomial numbers A000984(n) = (2*n)!/n!^2 have the property that 60*A000984(n) is divisible by (n + 1)*(n + 2)*(n + 3) and the result (2*n)!/(n!*(n+3)!) is the super ballot number A007272(n). Similarly, the numbers A008977(n) = (4*n)!/n!^4 appear to have the property that 9979200*A008977(n) is divisible by ((n + 1)*(n + 2)*(n + 3))^3, leading to the present sequence. Cf. A361030. %C A361035 Conjecture: a(n) is odd iff n = 2^k - 3 for some k >= 2. %F A361035 a(n) = 9979200 * A008977(n)/((n+1)*(n+2)*(n+3))^3. %F A361035 a(n) = (15925)*A008977(n+3)/((4*n+1)*(4*n+2)*(4*n+3)*(4*n+5)*(4*n+6)*(4*n+7)*(4*n+9)*(4*n+10)*(4*n+11)). %F A361035 P-recursive: a(n) = 4*(4*n-1)*(4*n-2)*(4*n-3)/(n+3)^3 * a(n-1) with a(0) = 46200. %F A361035 The o.g.f. A(x) satisfies the differential equation %F A361035 x^3*(1 - 256*x)*A(x)''' + x^2*(12 - 1152*x)*A(x)'' + x*(37 - 816*x)*A(x)' + (27 - 24*x)*A(x) - 1247400 = 0 with A(0) = 46200, A'(0) = 17325 and A''(0) = 232848. %F A361035 a(n) ~ 2494800*sqrt(8/Pi^3) * 2^(8*n)/n^(21/2). %p A361035 seq( 9979200 * (4*n)!/(n!*(n+3)!^3 ), n = 0..20); %Y A361035 Cf. A007272, A008977, A361030, A361032, A361033, A361034. %K A361035 nonn,easy %O A361035 0,1 %A A361035 _Peter Bala_, Mar 01 2023