A026029 Number of (s(0), s(1), ..., s(2n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 3, s(2n) = 3. Also T(2n,n), where T is defined in A026022.
1, 2, 6, 20, 69, 242, 858, 3068, 11050, 40052, 145996, 534888, 1968685, 7276050, 26993490, 100490220, 375287550, 1405622460, 5278838100, 19873977240, 74994427170, 283595947284, 1074568266756, 4079184055640, 15511924233204, 59083160374952, 225384613313944
Offset: 0
Keywords
Links
- Michael De Vlieger, 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.
- 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 - 2*x)*(-1 + Sqrt[1 - 4*x] + 2*x)^2 / (4*x^4), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 03 2019 *)
Formula
Expansion of (1+x^2*C^4)*C^2, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
a(n) = Sum_{k=0..n} C(n, k)*Sum_{i=0..k} C(k, 2i)*A000108(i+1). - Paul Barry, Jul 18 2003
a(n) = Sum_{k=0..3} A039599(n,k) = A000108(n) + A000245(n) + A000344(n) + A000588(n) = A026012(n) + A000588(n). - Philippe Deléham, Nov 12 2008
a(n) = C(2n,n) - C(2n,n-4). - Paul Barry, May 11 2009
Conjecture: (n+4)*a(n) + 6*(-n-2)*a(n-1) + 4*(2*n-1)*a(n-2) = 0. - R. J. Mathar, Nov 24 2012
a(n) ~ 4^(n+2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 03 2019
E.g.f.: exp(2*x)*(BesselI(0, 2*x) - BesselI(4, 2*x)). - Stefano Spezia, Jan 17 2024
Comments