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 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 ends with x, and P is applied n times in the word. 1
 1, 3, 12, 60, 336, 2016, 12672, 82368, 549120, 3734016, 25798656, 180590592, 1278025728, 9128755200, 65727037440, 476521021440, 3475800391680, 25489202872320, 187815179059200, 1389832325038080, 10324468700282880 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Indranil Ghosh, Table of n, a(n) for n = 0..1000 L. Guo and W. Sit, Enumeration and generating functions of Rota-Baxter words, 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). FORMULA a(n) = 3*2^(n-1)C(n), where C(n) is the n-th Catalan number. G.F. (3-4t-3 sqrt(1-8t))/(8t). (n+1)*a(n) = 4*(2*n-1)*a(n-1). - R. J. Mathar, Jul 24 2012 EXAMPLE 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 MATHEMATICA series[n_]:= Module[{}, If[n == 0, 1]; a = Series[(3 - 4 z - 3 Sqrt[1 - 8 z])/(8 z), {z, 0, n}]; Join[{1}, Coefficient[a, Table[z^i, {i, 1, n}]]]] a[0] = 1; a[n_]:= 3*2^(n-1) CatalanNumber[n]; Table[a[n], {n, 0, 20}] (* Indranil Ghosh, Mar 05 2017 *) PROG (PARI) a(n) = if(n==0, 1, 3*2^(n-1)*(binomial(2*n, n)/(n+1))); \\ Indranil Ghosh, Mar 05 2017 (Python) import math f = math.factorial def C(n, r): return f(n)/f(r)/f(n-r) def A181282(n): return 1 if n==0 else 3*2**(n-1)*(C(2*n, n)/(n+1)) # Indranil Ghosh, Mar 05 2017 CROSSREFS Cf. A003645, A025225 Sequence in context: A278395 A128602 A092803 * A020052 A096471 A140097 Adjacent sequences:  A181279 A181280 A181281 * A181283 A181284 A181285 KEYWORD nonn AUTHOR William Sit (wyscc(AT)sci.ccny.cuny.edu), Oct 11 2010 STATUS approved

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Last modified August 3 05:55 EDT 2020. Contains 336197 sequences. (Running on oeis4.)