A078482 Expansion of g.f. (1 - 3*x + x^2 - sqrt(1 - 6*x + 7*x^2 - 2*x^3 + x^4))/(2*x).
0, 1, 2, 6, 20, 70, 254, 948, 3618, 14058, 55432, 221262, 892346, 3630680, 14885042, 61432382, 255025212, 1064190214, 4461325382, 18780710508, 79357572866, 336466650450, 1431007889744, 6103431668830, 26099839562738, 111877997049648, 480635694869218
Offset: 0
Keywords
Examples
G.f.: A(x) = x + 2*x^2 + 6*x^3 + 20*x^4 + 70*x^5 + 254*x^6 + 948*x^7 + ... From _Paul D. Hanna_, Sep 12 2012: (Start) The logarithm of the g.f. begins log(A(x)/x) = (1 + 1/(1-x))*x + (1 + 2^2/(1-x) + 1/(1-x)^2)*x^2/2 + (1 + 3^2/(1-x) + 3^2/(1-x)^2 + 1/(1-x)^3)*x^3/3 + (1 + 4^2/(1-x) + 6^2/(1-x)^2 + 4^2/(1-x)^3 + 1/(1-x)^4)*x^4/4 + (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 + ... (End) 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
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Andrei Asinowski and Cyril Banderier, From geometry to generating functions: rectangulations and permutations, arXiv:2401.05558 [cs.DM], 2024. See page 2.
- Andrei Asinowski, Gill Barequet, Mireille Bousquet-Mélou, Toufik Mansour, and Ron Y. Pinter, Orders induced by segments in floorplan partitions and (2-14-3, 3-41-2)-avoiding permutations arXiv:1011.1889 [math.CO], 2010-2012, page 32.
- Andrei Asinowski and Toufik Mansour, Separable d-Permutations and Guillotine Partitions, arXiv 0803.3414 [math.CO], 2008. Annals of Combinatorics 14 (1) pp.17-43 Springer, 2010; Abstract
- M. D. Atkinson and T. Stitt, Restricted permutations and the wreath product, Preprint, 2002.
- M. D. Atkinson and T. Stitt, Restricted permutations and the wreath product, Discrete Math., 259 (2002), 19-36.
- Christian Bean, Émile Nadeau, and Henning Ulfarsson, Enumeration of Permutation Classes and Weighted Labelled Independent Sets, arXiv:1912.07503 [math.CO], 2019.
- CombOS - Combinatorial Object Server, Generate 1-sided guillotine rectangulations
- Elizabeth Hartung, Hung Phuc Hoang, Torsten Mütze, and Aaron Williams, Combinatorial generation via permutation languages. I. Fundamentals, arXiv:1906.06069 [cs.DM], 2019.
- Arturo Merino and Torsten Mütze, Combinatorial generation via permutation languages. III. Rectangulations, arXiv:2103.09333 [math.CO], 2021.
Programs
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Mathematica
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 *) -
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 */
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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 -
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
Formula
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
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
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
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
Extensions
Replaced definition with g.f. given by Atkinson and Still (2002). - N. J. A. Sloane, May 24 2016
Comments