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 A387250 #11 Aug 29 2025 11:41:28 %S A387250 105,735,50050,5460315,742511070,114872107350,19348562209860, %T A387250 3461691866723475,647897423565562310,125577883051534761666, %U A387250 25029394494457424675100,5103876046438721064520350,1060725331955983336553011500,224018752093294694626068131340,47967198494914114482847609250184 %N A387250 a(n) = 105/(n + 1) * Catalan(4*n). %C A387250 For r >= 2, there is a constant K_r such that K_r/(n + 1) * Catalan(r*n) is integral for all n. %F A387250 a(n) = 105/((n + 1)*(4*n + 1)) * binomial(8*n, 4*n). %F A387250 a(n) = 2*(8*n - 1)*(8*n - 3)*(8*n - 5)*(8*n - 7)/((n + 1)*(2*n - 1)*(4*n + 1)*(4*n - 1)) * a(n-1) with a(0) = 105. %F A387250 a(n) ~ 105/(8*sqrt(Pi)) * 256^n/n^(5/2). %F A387250 E.g.f.: 105*hypergeom([1/8, 3/8, 5/8, 7/8], [1/2, 3/4, 5/4, 2], 256*x). - _Stefano Spezia_, Aug 27 2025 %p A387250 seq( 105/((n + 1)*(4*n + 1)) * binomial(8*n, 4*n), n = 0..20); %t A387250 a[n_]:=105/(n+1)*CatalanNumber[4n];Array[a,15,0] (* _James C. McMahon_, Aug 29 2025 *) %Y A387250 Cf. A000108, A387248, A387249. %K A387250 nonn,easy,new %O A387250 0,1 %A A387250 _Peter Bala_, Aug 25 2025