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A106228 G.f. satisfies A(x) = 1 + x*A(x)/(1 - x*A(x)^2).

%I A106228 #193 Feb 24 2025 06:28:21
%S A106228 1,1,2,6,21,80,322,1347,5798,25512,114236,518848,2384538,11068567,
%T A106228 51817118,244370806,1159883685,5536508864,26560581688,127993221140,
%U A106228 619280193640,3007251366000,14651743202152,71601107803904,350873710447210,1723795243004223
%N A106228 G.f. satisfies A(x) = 1 + x*A(x)/(1 - x*A(x)^2).
%C A106228 Number of paths from (0,0) to (3n-3,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1) and have no tripledescents (ddd). Example: a(3)=6 because we have udud, Uddud, udUdd, UddUdd, uudd and Ududd (the remaining four paths contain the string ddd: uUddd, UdUddd, Uuddd and UUdddd; see A027307). - _Emeric Deutsch_, Jun 08 2005
%C A106228 a(n) = number of node-labeled ordered trees (A000108) on n vertices, each node labeled with a positive integer <= its outdegree. A node is a non-root non-leaf vertex. Example. a(3)=6 counts the 5 ordered trees on 4 vertices with all labels 1 and the tree
%C A106228 .|.
%C A106228 / \
%C A106228 with its (one and only) node labeled 2. - _David Callan_, Jul 14 2006
%C A106228 a(n) = number of Schroeder (n-1)-paths with no triple descents. Example: a(4)=21 counts all 22 Schroeder 3-paths (A006318) except UUUDDD. - _David Callan_, Jul 14 2006
%C A106228 (1 + 2x + 6x^2 + ...)*(1 + x + 2x^2 + 6x^3) = (1 + 3x + 10x^2 + 37x^3 + ...), where A109081 = (1, 1, 3, 10, 37, ...). - _Gary W. Adamson_, Nov 15 2011
%C A106228 a(n) = number of Motzkin paths of length 2n-1 with no downsteps in odd position. Example: a(3)=6 counts FFFFF, FFUDF, FUFDF, UDFFF, UDUDF, UFFDF with U an upstep (1,1), F a flatstep (1,0), and D a downstep (1,-1). - _David Callan_, May 20 2015
%C A106228 Number of permutations of length n that avoid 4123, 4132, and 4213. - _Jay Pantone_, Oct 01 2015
%C A106228 Conjecturally, the number of sequences (e(1), ..., e(n)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) > e(j) and e(i) <= e(k). [Martinez and Savage, 2.21] - _Eric M. Schmidt_, Jul 17 2017
%C A106228 a(n) is the number of permutations of length n avoiding the partially ordered pattern (POP) {1>3, 1>4, 4>2} of length 4. That is, the number of length n permutations having no subsequences of length 4 in which the first element is the largest and the fourth element is larger than the second element. - _Sergey Kitaev_, Dec 10 2020
%C A106228 a(n) is the number of peakless Motzkin paths of length 2n that do not start with an up edge and where every pair of matching up and down edges occupies positions of the same parity. Equivalently, the number of RNA secondary structures on 2n vertices where the leftmost vertex is not matched and only vertices of the same parity can be matched. - _Alexander Burstein_, May 17 2021
%H A106228 Jay Pantone, <a href="/A106228/b106228.txt">Table of n, a(n) for n = 0..500</a>
%H A106228 Michael H. Albert, Cheyne Homberger, Jay Pantone, Nathaniel Shar, and Vincent Vatter, <a href="http://arxiv.org/abs/1510.00269">Generating Permutations with Restricted Containers</a>, arXiv:1510.00269 [math.CO], 2015.
%H A106228 Andrei Asinowski and Cyril Banderier, <a href="https://arxiv.org/abs/2401.05558">From geometry to generating functions: rectangulations and permutations</a>, arXiv:2401.05558 [cs.DM], 2024. See page 2.
%H A106228 E. Barcucci, R. Pinzani and R. Sprugnoli, <a href="https://users.dimi.uniud.it/~giacomo.dellariccia/Table%20of%20contents/BarcucciPinzaniSprugnoli1991.pdf">The Motzkin family</a>, P.U.M.A. Ser. A, Vol. 2, 1991, No. 3-4, pp. 249-279.
%H A106228 Paul Barry, <a href="http://dx.doi.org/10.1016/j.laa.2015.10.032">Riordan arrays, generalized Narayana triangles, and series reversion</a>, Linear Algebra and its Applications, 491 (2016) 343-385.
%H A106228 N. R. Beaton, M. Bouvel, V. Guerrini, and S. Rinaldi, <a href="https://doi.org/10.37236/7375">Slicings of Parallelogram Polyominoes: Catalan, Schröder, Baxter, and Other Sequences</a>, Electr. J. Combin. 26 (2019), P3.13. Also at <a href="https://arxiv.org/abs/1511.04864">arXiv:1511.04854</a> [math.CO], 2015-2019.
%H A106228 Beáta Bényi, Toufik Mansour, and José L. Ramírez, <a href="https://arxiv.org/abs/2309.06518">Pattern Avoidance in Weak Ascent Sequences</a>, arXiv:2309.06518 [math.CO], 2023.
%H A106228 Wlodzimierz Bryc, Raouf Fakhfakh, and Wojciech Mlotkowski, <a href="https://arxiv.org/abs/1708.05343">Cauchy-Stieltjes families with polynomial variance functions and generalized orthogonality</a>, arXiv:1708.05343 [math.PR], 2017-2019. Also in <a href="http://www.math.uni.wroc.pl/~pms/publicationsArticleRef.php?nr=39.2&amp;nrA=1&amp;ppB=%20237&amp;ppE=%20258">Probability and Mathematical Statistics</a> (2019), Vol. 39, No. 2, 237-258.
%H A106228 Alexander Burstein and Louis W. Shapiro, <a href="https://arxiv.org/abs/2112.11595">Pseudo-involutions in the Riordan group</a>, arXiv:2112.11595 [math.CO], 2021.
%H A106228 David Callan and Toufik Mansour, <a href="https://doi.org/10.1515/puma-2015-0027">Enumeration of small Wilf classes avoiding 1342 and two other 4-letter patterns</a>, Pure Mathematics and Applications (2018) Vol. 27, No. 1, 62-97.
%H A106228 Wenqin Cao, Emma Yu Jin, and Zhicong Lin, <a href="https://doi.org/10.1016/j.dam.2019.01.035">Enumeration of inversion sequences avoiding triples of relations</a>, Discrete Applied Mathematics (2019); see also <a href="http://www.emmayujin.at/Pubs/CaoJinLin19.pdf">author's copy</a>
%H A106228 Emeric Deutsch, <a href="http://www.jstor.org/stable/2589192">Problem 10658</a>, American Math. Monthly, 107, 2000, 368-370.
%H A106228 Alice L. L. Gao and Sergey Kitaev, <a href="https://arxiv.org/abs/1903.08946">On partially ordered patterns of length 4 and 5 in permutations</a>, arXiv:1903.08946 [math.CO], 2019.
%H A106228 Alice L. L. Gao and Sergey Kitaev, <a href="https://doi.org/10.37236/8605">On partially ordered patterns of length 4 and 5 in permutations</a>, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
%H A106228 Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, <a href="http://dx.doi.org/10.1016/j.disc.2007.04.007">2-Binary trees: bijections and related issues</a>, Discr. Math., 308 (2008), 1209-1221.
%H A106228 JiSun Huh, Sangwook Kim, Seunghyun Seo, and Heesung Shin, <a href="https://arxiv.org/abs/2404.04091">Bijections on pattern avoiding inversion sequences and related objects</a>, arXiv:2404.04091 [math.CO], 2024. See pp. 2, 22.
%H A106228 Megan A. Martinez and Carla D. Savage, <a href="https://arxiv.org/abs/1609.08106">Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations</a>, arXiv:1609.08106 [math.CO], 2016.
%H A106228 Arturo Merino and Torsten Mütze, <a href="https://arxiv.org/abs/2103.09333">Combinatorial generation via permutation languages. III. Rectangulations</a>, arXiv:2103.09333 [math.CO], 2021.
%H A106228 Chunyan Yan and Zhicong Lin, <a href="https://arxiv.org/abs/1912.03674">Inversion sequences avoiding pairs of patterns</a>, arXiv:1912.03674 [math.CO], 2019.
%F A106228 G.f.: A(x) = (1/x)*series_reversion[x/(1 + x*G001006(x))] and thus G.f. satisfies: A(x) = 1 + x*A(x)*G001006(x*A(x)) where G001006(x) is the g.f. of Motzkin numbers A001006.
%F A106228 G.f.: 1 + x*exp( Sum_{n>=1} A082759(n)*x^n/n ), where A082759(n) = Sum_{k=0..n} binomial(n,k)*trinomial(n,k). - _Paul D. Hanna_, Nov 02 2012
%F A106228 a(n) = (1/n)sum(binomial(n, j+1)*b(n, j), j=0..n-1), where b(n, j) are the trinomial coefficients [b(n, j)=A027907(n, j)=coefficient of x^j in (1+x+x^2)^n]. - _Emeric Deutsch_, Jun 08 2005
%F A106228 Given g.f. A(x), then B(x) = x*A(x) satisfies 0 = f(x, B(x)) where f(x, y) = y^3 - (1+y)*x*(y-x). - _Michael Somos_, Jun 18 2005
%F A106228 a(n+1) = Sum[binomial(2n-2k,n-k)*binomial(n+k,n)/(n+1),{k,0,n}]. - _David Callan_, Aug 16 2006
%F A106228 For n>0: a(n) = 1/n*sum(binomial(n,j)*sum(binomial(j,i)*binomial(n-j,2*j-n-i-1)*2^(2*n-3*j+2*i+1),i=0..n-1), j=0..n); - _Vladimir Kruchinin_, Dec 26 2010
%F A106228 a(n) = 1/(n+1)*sum(binomial(n+1,k)*binomial(n+k+1,n-k),k,0,n); - _Vladimir Kruchinin_, Feb 28 2010
%F A106228 a(n) = upper left term in M^n, M = the production matrix:
%F A106228   1, 1
%F A106228   1, 1, 1
%F A106228   2, 2, 1, 1
%F A106228   3, 3, 2, 1, 1
%F A106228   4, 4, 3, 2, 1, 1
%F A106228   5, 5, 4, 3, 2, 1, 1
%F A106228   ...
%F A106228 - _Gary W. Adamson_, Jul 08 2011
%F A106228 D-finite with recurrence: 4*n*(2*n+1)*a(n) + 2*(6-5*n-10*n^2)*a(n-1) + 12*(-9*n^2+35*n-33)*a(n-2) - 2*(n-3)*(13*n-28)*a(n-3) - 15*(n-3)*(n-4)*a(n-4) = 0. - _R. J. Mathar_, Nov 14 2011
%F A106228 From _Gary W. Adamson_, Nov 15 2011: (Start)
%F A106228 a(n) is the sum of top row terms of Q^(n-1), where Q = the following infinite square production matrix:
%F A106228   1, 1, 0, 0, 0, ...
%F A106228   2, 1, 1, 0, 0, ...
%F A106228   3, 2, 1, 1, 0, ...
%F A106228   4, 3, 2, 1, 1, ...
%F A106228   5, 4, 3, 2, 1, ...
%F A106228   ... (End)
%F A106228 a(n) = 3_F_2([-n, 1-n, n+1], [1, 3/2], 1/4). - _Peter Luschny_, Aug 02 2012
%F A106228 A four-term recurrence equation is given in the Maple program. _Peter Luschny_, Aug 03 2012
%F A106228 a(n) ~ 1/228*sqrt(114)*sqrt((32129+3933*sqrt(57))^(1/3) * ((32129+3933*sqrt(57))^(2/3) + 532 + 38*(32129+3933*sqrt(57))^(1/3))) / ((32129+3933*sqrt(57))^(1/3)) * (((1261+57*sqrt(57))^(2/3) + 112 + 10*(1261+57*sqrt(57))^(1/3)) / (6*(1261+57*sqrt(57))^(1/3)))^n / (sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Sep 16 2013
%F A106228 G.f. satisfies x*F(x)^3 - x*F(x)^2 + (x-1)*F(x) + 1 = 0. - _Jay Pantone_, Oct 01 2015
%F A106228 G.f. satisfies A(-x*A(x)^3) = 1/A(x). - _Alexander Burstein_, Dec 05 2019
%F A106228 G.f.: A(x) = sqrt(B(x)) where B(x) is the g.f. of A262441. - _Seiichi Manyama_, Mar 31 2024
%F A106228 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n/2+3*k/2+1/2,n)/(n+3*k+1). - _Seiichi Manyama_, Apr 04 2024
%e A106228 A = 1 + x*A + x^2*A^3 + x^3*A^5 + x^4*A^7 + x^5*A^9 + ...
%e A106228 a(4) = 21 since the top row terms of Q^3 = (10, 7, 3, 1). - _Gary W. Adamson_, Nov 15 2011
%e A106228 G.f. = 1 + x + 2*x^2 + 6*x^3 + 21*x^4 + 80*x^5 + 322*x^6 + 1347*x^7 + ...
%p A106228 a := proc(n) option remember; if n<2 then 1 elif n=2 then 2 else ((380*n^3-840*n^2+496*n-72)*a(n-1)+(76*n^3-282*n^2+302*n-84)*a(n-2)+(57*n^3-297*n^2+402*n-72)*a(n-3))/(76*n^3-54*n^2-46*n) fi end: seq(a(i),i=0..23); # _Peter Luschny_, Aug 03 2012
%t A106228 Flatten[{1,Table[1/n*Sum[Binomial[n,k]*Binomial[n+k,n-k-1],{k,0,n-1}],{n,1,20}]}] (* _Vaclav Kotesovec_, Sep 16 2013 *)
%t A106228 a[ n_] := If[ n < 0, 0, HypergeometricPFQ[{-n, 1 - n, n + 1}, {1, 3/2}, 1/4]]; (* _Michael Somos_, May 27 2014 *)
%o A106228 (PARI) {a(n)=local(A=1+x+x*O(x^n));for(k=1,n,A=1+x*A/(1-x*A^2)); polcoeff(A,n)}
%o A106228 (PARI) {a(n) = local(A); if( n<0, 0, n++; A = (1 + x - sqrt(1 - 2*x - 3*x^2 + x * O(x^n))) / 2; polcoeff( serreverse( x^2 / A), n))}; /* _Michael Somos_, Jun 18 2005 */
%o A106228 (Sage)
%o A106228 from mpmath import mp
%o A106228 mp.dps = 32; mp.pretty = True
%o A106228 def A106228(n) : return int(mp.hyper([-n, 1-n, n+1], [1, 3/2], 1/4))
%o A106228 [A106228(n) for n in (0..23)] # _Peter Luschny_, Aug 02 2012
%o A106228 (PARI)  {a(n) = if( n<1, n==0, polcoeff( serreverse( x /  (1 + 2*x + 2*x^2 + x^3) + x * O(x^n)), n))}; /* _Michael Somos_, Dec 31 2014 */
%Y A106228 Cf. A027907, A027307, A001006, A086246, A109081, A164965, A262441.
%K A106228 nonn
%O A106228 0,3
%A A106228 _Paul D. Hanna_, May 19 2005