|
|
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.
|
|
2
|
|
|
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
|
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
|
G.f.: (3 - 4*t - 3*sqrt(1-8*t))/(8*t).
|
|
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)
(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
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
William Sit (wyscc(AT)sci.ccny.cuny.edu), Oct 11 2010
|
|
STATUS
|
approved
|
|
|
|