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.

1, 3, 12, 60, 336, 2016, 12672, 82368, 549120, 3734016, 25798656, 180590592, 1278025728, 9128755200, 65727037440, 476521021440, 3475800391680, 25489202872320, 187815179059200, 1389832325038080, 10324468700282880

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)*A000108(n).

G.f.: (3 - 4*t - 3*sqrt(1-8*t))/(8*t).

(n+1)*a(n) = 4*(2*n-1)*a(n-1). - R. J. Mathar, Jul 24 2012

a(n) = (n+2) * A000257(n). - F. Chapoton, Feb 26 2024

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

CoefficientList[Series[(3-4x-3Sqrt[1-8x])/(8x), {x, 0, 40}], x]
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
(Magma) [1] cat [3*2^(n-1)*Catalan(n): n in [1..40]]; // G. C. Greubel, Jan 04 2023
(SageMath) [3*2^(n-1)*catalan_number(n) -int(n==0)/2 for n in range(41)] # G. C. Greubel, Jan 04 2023

Crossrefs

Cf. A000108, A003645, A025225, A000257.

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