OFFSET
0,3
COMMENTS
A minor variation of A006318. Unsigned version of A086456 and A103137. The Hankel transform of this sequence is A006125.
a(n) is also the number of "branching configurations" for RNA (see Sankoff, 1985) that have exactly n hairpins. - Lee A. Newberg, Mar 30 2010
a(n) is also the number of ways to insert balanced parentheses into a product of n variables such that each parenthesis pair has 2 or more top-level factors. - Lee A. Newberg, Apr 06 2010
a(n) is also the number of infix expressions with n variables and operators + and - such that there are no redundant parentheses. - Vjeran Crnjak, Apr 25 2020
a(n) is also the number of permutations on n elements that can be obtained with an output-restricted (or input-restricted) deque. (see D. E. Knuth: The Art of Computer Programming, Volume 1, page 539). - Zhujun Zhang, Oct 15 2023
REFERENCES
S. Kitaev, Patterns in Permutations and Words, Springer-Verlag, 2011. see p. 399 Table A.7
D. E. Knuth, The Art of Computer Programming, Volume 1, Fundamental Algorithms, section 2.2.1: Stacks, Queues, and Deques.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..1313
J. Abate and W. Whitt, Integer Sequences from Queueing Theory , J. Int. Seq. 13 (2010), 10.5.5, Theorem 5.
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
Christian Bean, Finding structure in permutation sets, Ph.D. Dissertation, Reykjavík University, School of Computer Science, 2018.
Arnauld Mesinga Mwafise, Computational and Combinatorial Enumeration of Poset Matrices, 2024. See p. 8.
D. Sankoff, Simultaneous solution of the RNA folding, alignment and protosequence problems, Siam J. Appl. Math 45(5):810-825 (1985). [From Lee A. Newberg, Mar 30 2010]
FORMULA
G.f.: (3 - x - sqrt(1 -6*x +x^2))/2.
G.f.: 4 / (3 - x + sqrt(1 - 6*x + x^2)). - Michael Somos, Apr 18 2012
a(n) ~ sqrt((sqrt(18)-4)/(4*Pi)) * n^(-3/2) * (3 + sqrt(8))^n, which is, approximately, a(n) ~ 0.1389558648 * n^(-1.5) * 5.828427125^n. - Lee A. Newberg, Apr 06 2010
a(n) ~ (1 + sqrt(2))^(2*n-1) / (2^(3/4)*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 23 2023
a(n) = top left term of M^n, where M = the production matrix:
1, 1, 0, 0, 0, ...
1, 2, 1, 0, 0, ...
1, 2, 2, 1, 0, ...
1, 2, 2, 2, 1, ...
1, 2, 2, 2, 2, 1, ...
...
Top row terms of M^n generates rows of triangle A132372. - Gary W. Adamson, Jul 07 2011
G.f.: A(x)=(3 -x- sqrt(1-6*x+x^2))/2= 2 - G(0); G(k)= 1 + x - 2*x/G(k+1); (continued fraction, 1-step, 1 var.). - Sergei N. Gladkovskii, Jan 04 2012
G.f.: A(x)=(3 -x -sqrt(1-6*x+x^2))/2= G(0); G(k)= := 1 - x/(1 - 2/G(k+1)); (continued fraction, 2-step, 2 var.). - Sergei N. Gladkovskii, Jan 04 2012
D-finite with recurrence: n*a(n) +3*(3-2*n)*a(n-1) +(n-3)*a(n-2)=0. - R. J. Mathar, Jul 24 2012
G.f.: 1 / (1 - x / (1 - x / (1 - 2*x / (1 - x / (1 - 2*x / (1 - x / ... )))))) = 1 + x / (1 - 2*x / (1 - x / (1 - 2*x / (1 - x / (1 - 2*x / (1 - x / ... )))))). - Michael Somos, Jan 03 2013
G.f.: 2 - x - G(0), where G(k)= k+1 - 2*x*(k+1) - 2*x*(k+1)*(k+2)/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Jul 14 2013
a(n) = (1/n)*Sum_{i = 0..floor(n/2)} binomial(n+i-1, i)*binomial(2*n, n-2*i-1), n>0, a(0)=1. - Vladimir Kruchinin, Nov 13 2014
a(n) = Catalan(n)*hypergeometric([1/2-n/2, 1-n/2, n], [n/2+1, n/2+3/2], 1). - Peter Luschny, Nov 14 2014
a(0) = 1; a(n) = a(n-1) + Sum_{k=1..n-1} a(k) * a(n-k). - Ilya Gutkovskiy, Apr 11 2021
EXAMPLE
From Lee A. Newberg, Mar 30 2010: (Start)
For n = 2, the a(2) = 2 branching configurations are ()() and (()()), where each () indicates a hairpin (also termed 1-loop) and each other pair of parentheses indicates a k-loop for k >= 3.
For n = 3, the a(3) = 6 branching configurations are ()()(), (()())(), ()(()()), (()()()), ((()())()), and (()(()())). (End)
When inserting balanced parentheses into the product x^n: For n = 0, the a(0) = 1 possible term is the empty term. For n = 1, the a(1) = 1 possible term is x. For n = 2, the a(2) = 2 possible terms are xx and (xx). For n = 3, the a(3) = 6 possible terms are xxx, (xx)x, x(xx), (xxx), ((xx)x), and (x(xx)). - Lee A. Newberg, Apr 06 2010
G.f. = 1 + x + 2*x^2 + 6*x^3 + 22*x^4 + 90*x^5 + 394*x^6 + 1806*x^7 + ...
MAPLE
seq(coeff(series((3-x -sqrt(1-6*x+x^2))/2, x, n+1), x, n), n = 0..25); # G. C. Greubel, Jun 08 2020
MATHEMATICA
CoefficientList[Series[(3 -x -Sqrt[1-6x+x^2])/2, {x, 0, 25}], x] (* Vincenzo Librandi, Nov 13 2014 *)
PROG
(Maxima)
a(n):=if n<1 then 1 else sum(binomial(n+i-1, i)* binomial(2*n, n-2*i-1), i, 0, (n)/2)/(n); /* Vladimir Kruchinin, Nov 13 2014 */
(Sage)
a = lambda n: catalan_number(n)*hypergeometric([1/2-n/2, 1-n/2, n], [n/2+1, n/2+3/2], 1)
print([simplify(a(n)) for n in (0..25)]) # Peter Luschny, Nov 14 2014
(Magma) R<x>:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (3-x-Sqrt(1-6*x+x^2))/2 )); // G. C. Greubel, Jun 08 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Philippe Deléham, Nov 02 2009
STATUS
approved