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a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 2, s(n) = 4. Also a(n) = T(n,n-2), where T is the array in A026323.
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%I #17 Feb 08 2017 02:08:19

%S 1,3,10,30,90,266,783,2295,6710,19580,57057,166101,483210,1405080,

%T 4084590,11872494,34508997,100313635,291646580,848102640,2466916474,

%U 7177785582,20891443950,60827142350,177167486925,516217883571,1504692189588

%N a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 2, s(n) = 4. Also a(n) = T(n,n-2), where T is the array in A026323.

%C Number of paths in the plane x>=0 and y>=-2, from (0,0) to (n,2), and consisting of steps U=(1,1), D=(1,-1) and H=(1,0). For example, for n=4, we have the 10 paths: UUUD, UUHH, UUDU, UHUH, UHHU, UDUU, HUUH, HUHU, HHUU, DUUU. - _José Luis Ramírez Ramírez_, Apr 20 2015

%H G. C. Greubel, <a href="/A026327/b026327.txt">Table of n, a(n) for n = 2..1000</a>

%F Conjecture: -(n+8)*(n-2)*a(n) +3*(2*n^2+7*n-28)*a(n-1) +3*(-3*n^2-n+36)*a(n-2) -4*(n+4)*(n-1)*a(n-3) +12*(n-1)*(n-2)*a(n-4)=0. - _R. J. Mathar_, Jun 23 2013

%F G.f: x^2*M(x)^2/(1-x-x^2*(M(x)+1/(1-x-x^2/(1-x)))), where M(x) is g.f. of Motzkin paths. - _José Luis Ramírez Ramírez_, Apr 20 2015

%F a(n) ~ 5 * 3^(n+5/2) / (2*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Apr 21 2015

%t Rest[Rest[CoefficientList[Series[x^2*((1-x-Sqrt[1-2*x-3*x^2])/(2*x^2))^2/(1-x-x^2*((1-x-Sqrt[1-2*x-3*x^2])/(2*x^2)+1/(1-x-x^2/(1-x)))), {x, 0, 20}], x]]] (* _Vaclav Kotesovec_, Apr 21 2015 *)

%K nonn

%O 2,2

%A _Clark Kimberling_