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 A181282 #20 Jul 21 2024 16:35:41
%S A181282 1,3,12,60,336,2016,12672,82368,549120,3734016,25798656,180590592,
%T A181282 1278025728,9128755200,65727037440,476521021440,3475800391680,
%U A181282 25489202872320,187815179059200,1389832325038080,10324468700282880
%N A181282 a(n) is the number of associate Rota-Baxter words in one idempotent generator x and one idempotent operator P of degree n. Such words are Rota-Baxter words that begin and/or end with x, and P is applied n times in the word.
%H A181282 Indranil Ghosh, <a href="/A181282/b181282.txt">Table of n, a(n) for n = 0..1000</a>
%H A181282 L. Guo and W. Sit, <a href="http://dx.doi.org/10.1007/s11786-010-0061-2">Enumeration and generating functions of Rota-Baxter words</a>, Math. Comp. Sci. (2010). In G. Regensburger, M. Rosenkranz, and W. Sit, eds., Algebraic and Algorithmic Aspects of Differential and Integral Operators (AADIOS), Sp. Issue, Math. C. Sc., 4 (2,3) (2010).
%F A181282 a(n) = 3*2^(n-1)*A000108(n).
%F A181282 G.f.: (3 - 4*t - 3*sqrt(1-8*t))/(8*t).
%F A181282 (n+1)*a(n) = 4*(2*n-1)*a(n-1). - _R. J. Mathar_, Jul 24 2012
%F A181282 a(n) = (n+2) * A000257(n). - _F. Chapoton_, Feb 26 2024
%e A181282 For n = 2, the a(2) = 12 associate Rota-Baxter words are: xP(xP(x)), xP(xP(x))x, P(xP(x))x, xP(P(x)x), xP(P(x)x)x, P(P(x)x)x, xP(xP(x)x), xP(xP(x)x)x, P(xP(x)x)x, xP(x)xP(x), xP(x)xP(x)x, P(x)xP(x)x.
%t A181282 CoefficientList[Series[(3-4x-3Sqrt[1-8x])/(8x), {x,0,40}], x]
%t A181282 a[0] = 1; a[n_]:= 3*2^(n-1) CatalanNumber[n]; Table[a[n], {n,0,20}] (* _Indranil Ghosh_, Mar 05 2017 *)
%o A181282 (PARI) a(n) = if(n==0, 1, 3*2^(n-1)*(binomial(2*n,n)/(n+1))); \\ _Indranil Ghosh_, Mar 05 2017
%o A181282 (Python)
%o A181282 import math
%o A181282 f = math.factorial
%o A181282 def C(n,r): return f(n)/f(r)/f(n-r)
%o A181282 def A181282(n): return 1 if n==0 else 3*2**(n-1)*(C(2*n,n)/(n+1)) # _Indranil Ghosh_, Mar 05 2017
%o A181282 (Magma) [1] cat [3*2^(n-1)*Catalan(n): n in [1..40]]; // _G. C. Greubel_, Jan 04 2023
%o A181282 (SageMath) [3*2^(n-1)*catalan_number(n) -int(n==0)/2 for n in range(41)] # _G. C. Greubel_, Jan 04 2023
%Y A181282 Cf. A000108, A003645, A025225, A000257.
%K A181282 nonn
%O A181282 0,2
%A A181282 William Sit (wyscc(AT)sci.ccny.cuny.edu), Oct 11 2010