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%I #30 Jan 25 2023 10:05:22
%S 1,4,10,29,81,231,659,1891,5443,15718,45508,132067,384047,1118820,
%T 3264642,9539787,27913083,81769236,239794422,703906719,2068153899,
%U 6081507831,17896695831,52703944965,155310270101,457956633826,1351132539604
%N a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is an integer, s(0) = 0, |s(1)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2, s(n) = 1. Also a(n) = T(n,n-1), where T is the array defined in A025177.
%H G. C. Greubel, <a href="/A025179/b025179.txt">Table of n, a(n) for n = 2..1000</a> (terms 2..200 from T. D. Noe)
%F Equals (1/2) * A024997(n+1).
%F From _Vladeta Jovovic_, Jan 01 2004: (Start)
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*binomial(2*k+1, k+1).
%F E.g.f.: exp(x)*(BesselI(0, 2*x)+BesselI(2, 2*x)). (End)
%F From _Paul Barry_, Sep 17 2005: (Start)
%F G.f.: ((1-x)^2 - (1-x)*sqrt(1-2*x-3*x^2))/(2*x*sqrt(1-2*x-3*x^2)).
%F a(n+1) = Sum_{k=0..n} C(n, k)*C(k+1, k/2+1)*(1+(-1)^k)/2}. (End)
%F D-finite with recurrence (n+1)*a(n) +(-3*n+1)*a(n-1) +(-n-5)*a(n-2) +3*(n-3)*a(n-3)=0. - _R. J. Mathar_, Nov 26 2012
%F a(n) ~ 3^(n-1/2) / sqrt(Pi*n). - _Vaclav Kotesovec_, Feb 13 2014
%F Prepend 1 to the data, assume offset 0, and denote the resulting sequence alpha. Then alpha(n) = Sum_{k=0..n} Sum_{j=0..k} A359364(n, n - j). - _Peter Luschny_, Jan 10 2023
%t Rest[Rest[CoefficientList[Series[((1-x)^2-(1-x)*Sqrt[1-2*x-3*x^2]) /(2*x*Sqrt[1-2*x-3*x^2]), {x, 0, 20}], x]]] (* _Vaclav Kotesovec_, Feb 13 2014 *)
%o (PARI) x='x +O('x^50); Vec(((1-x)^2-(1-x +2*x^2)*sqrt(1-2*x-3*x^2)) /(2*x*sqrt(1 - 2*x -3*x^2))) \\ _G. C. Greubel_, Mar 01 2017
%Y Cf. A024997, A026135, A359364.
%K nonn
%O 2,2
%A _Clark Kimberling_
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