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 A361041 #8 Mar 07 2023 11:54:28 %S A361041 70,14,15,28,70,210,714,2660,10626,44850,197925,906192,4279240, %T A361041 20746936,102898110,520543380,2679559018,14007652050,74240555865, %U A361041 398363958300,2161524522150,11847660496770,65540249556600,365634339159024 %N A361041 a(n) = 1680*(3*n)!/(n!*(2*n + 4)!). %C A361041 Compare with the super ballot numbers A348893(n) = 840*(2*n)!/(n!*(n+4)!). %F A361041 a(n) = 70*binomial(3*n,2*n) - 196*binomial(3*n,2*n+1) + 141*binomial(3*n,2*n+2) - 65*binomial(3*n,2*n+3) + 14*binomial(3*n,2*n+4). Thus a(n) is an integer. %F A361041 P-recursive: 2*(n + 2)*(2*n + 3)*a(n) = 3*(3*n - 1)*(3*n - 2)*a(n-1) with a(0) = 70. %F A361041 a(n) ~ (27/4)^n * 105*sqrt(3/(4*Pi))/n^(9/2). %F A361041 The o.g.f. A(x) satisfies the differential equation %F A361041 x^2*(4 - 27*x^4)*A''(x) + 2*x*(9 - 27*x)*A'(x) + (12 - 6*x)*A(x) - 840 = 0, with A(0) = 70 and A'(0) = 14. %p A361041 seq( 1680*(3*n)!/(n!*(2*n + 4)!), n = 0..20); %Y A361041 Cf. A000139, A001764, A005809, A348893, A361039, A361040. %K A361041 nonn,easy %O A361041 0,1 %A A361041 _Peter Bala_, Mar 04 2023