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 A155069 Expansion of (3 - x - sqrt(1 - 6*x + x^2))/2. 7
 1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038, 3937603038, 20927156706, 111818026018, 600318853926, 3236724317174, 17518619320890, 95149655201962, 518431875418926 (list; graph; refs; listen; history; text; internal format)
 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 REFERENCES S. Kitaev, Patterns in Permutations and Words, Springer-Verlag, 2011. see p. 399 Table A.7 LINKS Michael De Vlieger, Table of n, a(n) for n = 0..1313 J. Abate, 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. 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) ~ 0.1389558645 * n^(-1.5) * 5.828427099^n. - Lee A. Newberg, Apr 06 2010 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..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:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (3-x-Sqrt(1-6*x+x^2))/2 )); // G. C. Greubel, Jun 08 2020 CROSSREFS Cf. A006318, A085403, A086456, A103137, A132372. Sequence in context: A049126 A049134 A086456 * A006318 A103137 A340892 Adjacent sequences:  A155066 A155067 A155068 * A155070 A155071 A155072 KEYWORD nonn AUTHOR Philippe Deléham, Nov 02 2009 STATUS approved

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Last modified October 4 12:02 EDT 2022. Contains 357239 sequences. (Running on oeis4.)