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 A386414 #19 Aug 01 2025 16:39:55 %S A386414 1,6,99,2142,52785,1404702,39331656,1141839504,34057559052, %T A386414 1037385419400,32133013365915,1009060082062110,32050934711814915, %U A386414 1027914968037080970,33240367148212098900,1082645830435810233960,35483717092533680418039,1169426742892003447650666 %N A386414 G.f. A(x) satisfies A(x) = (1 + 9*x*A(x)^3)^(2/3). %H A386414 Paolo Xausa, <a href="/A386414/b386414.txt">Table of n, a(n) for n = 0..600</a> %H A386414 Wikipedia, <a href="https://en.wikipedia.org/wiki/Fuss-Catalan_number">Fuss-Catalan number</a> %F A386414 a(n) = 9^n * binomial((6*n+2)/3,n)/(3*n+1). %F A386414 G.f.: B(x)^2, where B(x) is the g.f. of A008931. %F A386414 D-finite with recurrence +n*(3*n+2)*a(n) -6*(6*n-1)*(3*n-2)*a(n-1)=0. - _R. J. Mathar_, Jul 30 2025 %F A386414 G.f.: 2F1(1/3,5/6 ; 5/3 ; 36*x). - _R. J. Mathar_, Jul 30 2025 %t A386414 A386414[n_] := 9^n*Binomial[(6*n + 2)/3, n]/(3*n + 1); %t A386414 Array[A386414, 20, 0] (* _Paolo Xausa_, Aug 01 2025 *) %o A386414 (PARI) apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r); %o A386414 a(n) = 9^n*apr(n, 2, 2/3); %Y A386414 Cf. A004989, A377269, A386413, A386415. %Y A386414 Cf. A004987, A008931. %K A386414 nonn,easy %O A386414 0,2 %A A386414 _Seiichi Manyama_, Jul 21 2025