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A078482 Expansion of g.f. (1 - 3*x + x^2 - sqrt(1 - 6*x + 7*x^2 - 2*x^3 + x^4))/(2*x).

%I A078482 #93 Jan 11 2025 17:42:59
%S A078482 0,1,2,6,20,70,254,948,3618,14058,55432,221262,892346,3630680,
%T A078482 14885042,61432382,255025212,1064190214,4461325382,18780710508,
%U A078482 79357572866,336466650450,1431007889744,6103431668830,26099839562738,111877997049648,480635694869218
%N A078482 Expansion of g.f. (1 - 3*x + x^2 - sqrt(1 - 6*x + 7*x^2 - 2*x^3 + x^4))/(2*x).
%C A078482 Number of data structures of a certain wreath product type.
%C A078482 a(n) is also the number of (2-14-3, 3-41-2, 2-4-1-3, 3-1-4-2)-avoiding permutations. - _Mireille Bousquet-Mélou_, Jul 13 2012
%C A078482 Number of permutations that are separable by a point. And also the number of rectangulations that are guillotine and one-sided. - _Manfred Scheucher_, May 24 2023
%H A078482 Vincenzo Librandi, <a href="/A078482/b078482.txt">Table of n, a(n) for n = 0..1000</a>
%H A078482 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 A078482 Andrei Asinowski, Gill Barequet, Mireille Bousquet-Mélou, Toufik Mansour, and Ron Y. Pinter, <a href="https://arxiv.org/abs/1011.1889">Orders induced by segments in floorplan partitions and (2-14-3, 3-41-2)-avoiding permutations</a> arXiv:1011.1889 [math.CO], 2010-2012, page 32.
%H A078482 Andrei Asinowski and Toufik Mansour, <a href="http://arxiv.org/abs/0803.3414">Separable d-Permutations and Guillotine Partitions</a>, arXiv 0803.3414 [math.CO], 2008. Annals of Combinatorics 14 (1) pp.17-43 Springer, 2010; <a href="http://www.combinatorics.net/Annals/Abstract/14_1_17.aspx">Abstract</a>
%H A078482 M. D. Atkinson and T. Stitt, <a href="http://www.cs.otago.ac.nz/staffpriv/mike/Papers/WreathProduct/Wreathpaper.pdf">Restricted permutations and the wreath product</a>, Preprint, 2002.
%H A078482 M. D. Atkinson and T. Stitt, <a href="http://dx.doi.org/10.1016/S0012-365X(02)00443-0">Restricted permutations and the wreath product</a>, Discrete Math., 259 (2002), 19-36.
%H A078482 Christian Bean, Émile Nadeau, and Henning Ulfarsson, <a href="https://arxiv.org/abs/1912.07503">Enumeration of Permutation Classes and Weighted Labelled Independent Sets</a>, arXiv:1912.07503 [math.CO], 2019.
%H A078482 CombOS - Combinatorial Object Server, <a href="http://combos.org/rect">Generate 1-sided guillotine rectangulations</a>
%H A078482 Elizabeth Hartung, Hung Phuc Hoang, Torsten Mütze, and Aaron Williams, <a href="https://arxiv.org/abs/1906.06069">Combinatorial generation via permutation languages. I. Fundamentals</a>, arXiv:1906.06069 [cs.DM], 2019.
%H A078482 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.
%F A078482 a(n) = Sum_{m=1..n+1} C(m,n-m+1)*(Sum_{i=0..m-1} C(m,i)*C(2*m-i-2,m-1))*(-1)^(n-m+1)/m, n>0, a(0)=0. - _Vladimir Kruchinin_, May 21 2011
%F A078482 n*(n+1)*a(n) - 3*n*(2n-1)*a(n-1) + 7*n*(n-2)*a(n-2) - n*(2n-7)*a(n-3) + n*(n-5)*a(n-4) = 0. - _R. J. Mathar_, Jul 08 2012
%F A078482 G.f. satisfies: A(x) = x*(1 + A(x)) * (1 + A(x)/(1-x)). G.f.: x*exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 / (1-x)^k ). - _Paul D. Hanna_, Sep 12 2012
%F A078482 a(n) ~ sqrt(11*sqrt(2) - 16 + sqrt(16*sqrt(2) - 22)) * 2^(n - 1/2) / (sqrt(Pi) * n^(3/2) * (1 - sqrt(8*sqrt(2) - 11))^(n+1)). - _Vaclav Kotesovec_, May 17 2024
%e A078482 G.f.: A(x) = x + 2*x^2 + 6*x^3 + 20*x^4 + 70*x^5 + 254*x^6 + 948*x^7 + ...
%e A078482 From _Paul D. Hanna_, Sep 12 2012: (Start)
%e A078482 The logarithm of the g.f. begins
%e A078482 log(A(x)/x) = (1 + 1/(1-x))*x + (1 + 2^2/(1-x) + 1/(1-x)^2)*x^2/2 +
%e A078482 (1 + 3^2/(1-x) + 3^2/(1-x)^2 + 1/(1-x)^3)*x^3/3 +
%e A078482 (1 + 4^2/(1-x) + 6^2/(1-x)^2 + 4^2/(1-x)^3 + 1/(1-x)^4)*x^4/4 +
%e A078482 (1 + 5^2/(1-x) + 10^2/(1-x)^2 + 10^2/(1-x)^3 + 5^2/(1-x)^4 + 1/(1-x)^5)*x^5/5 + ...
%e A078482 (End)
%e A078482 a(5) = 70 = (1, 1, 2, 6, 20) dot product (1, 1, 3, 9, 29) = (29 + 9 + 6 + 6 + 20). - _Gary W. Adamson_, May 20 2013
%t A078482 CoefficientList[Series[(1 - 3 x + x^2 - Sqrt[1 - 6 x + 7 x^2 - 2 x^3 + x^4]) / (2 x), {x, 0, 40}], x] (* _Vincenzo Librandi_, May 28 2016 *)
%o A078482 (Maxima) a(n):=if n=0 then 0 else sum((binomial(m,n-m+1)* (sum(binomial(m,i)* binomial(2*m-i-2,m-1),i,0,m-1)) *(-1)^(n-m+1))/m,m,1,n+1); /* _Vladimir Kruchinin_, May 21 2011 */
%o A078482 (PARI) {a(n)=local(A=x);for(i=1,n,A=x*(1+A)*(1+A/(1-x +x*O(x^n))));polcoeff(A,n)} \\ _Paul D. Hanna_, Sep 12 2012
%o A078482 (PARI) {a(n)=polcoeff(x*exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^2/(1-x +x*O(x^n))^k))),n)} \\ _Paul D. Hanna_, Sep 12 2012
%Y A078482 Cf. A006318 (separable permutations), A078483.
%K A078482 nonn
%O A078482 0,3
%A A078482 _N. J. A. Sloane_, Jan 04 2003
%E A078482 Replaced definition with g.f. given by Atkinson and Still (2002). - _N. J. A. Sloane_, May 24 2016