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%I #35 Jan 17 2024 01:33:01
%S 1,2,6,20,69,242,858,3068,11050,40052,145996,534888,1968685,7276050,
%T 26993490,100490220,375287550,1405622460,5278838100,19873977240,
%U 74994427170,283595947284,1074568266756,4079184055640,15511924233204,59083160374952,225384613313944
%N 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.
%C Hankel transform is A008619(n+1). - _Paul Barry_, May 11 2009
%H Michael De Vlieger, <a href="/A026029/b026029.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 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 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.
%F a(n) = Sum_{k=0..n} C(n, k)*Sum_{i=0..k} C(k, 2i)*A000108(i+1). - _Paul Barry_, Jul 18 2003
%F 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
%F a(n) = C(2n,n) - C(2n,n-4). - _Paul Barry_, May 11 2009
%F 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
%F a(n) ~ 4^(n+2) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Sep 03 2019
%F E.g.f.: exp(2*x)*(BesselI(0, 2*x) - BesselI(4, 2*x)). - _Stefano Spezia_, Jan 17 2024
%t 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 *)
%K nonn
%O 0,2
%A _Clark Kimberling_
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