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%I #24 Mar 06 2017 02:41:11
%S 2,8,20,58,162,462,1318,3782,10886,31436,91016,264134,768094,2237640,
%T 6529284,19079574,55826166,163538472,479588844,1407813438,4136307798,
%U 12163015662,35793391662,105407889930,310620540202,915913267652,2702265079208
%N a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0 = s(n), |s(i) - s(i-1)| = 1 for i = 1,2; |s(i) - s(i-1)| <= 1 for i >= 3. Also a(n) = T(n,n), where T is the array defined in A024996.
%C Second differences of the central trinomial coefficients A002426. - _T. D. Noe_, Mar 16 2005
%H G. C. Greubel, <a href="/A024997/b024997.txt">Table of n, a(n) for n = 3..1000</a>
%F a(n) = 2*A025179(n-1).
%F From _G. C. Greubel_, Mar 01 2017: (Start)
%F a(n) = 2*Sum_{k=0..floor(n/2)} binomial(n, 2*k)*binomial(2*k+1, k+1), for n>=1.
%F O.g.f.: ((1-x)^2-(1-x+2*x^2)*sqrt(1-2*x-3*x^2)) / sqrt(1-2*x-3*x^2) [corrected by _Charles R Greathouse IV_, Mar 05 2017]
%F E.g.f.: 2*exp(x)*(BesselI(0, 2*x) + BesselI(2, 2*x)). (End)
%t Rest[Differences[CoefficientList[Series[1/Sqrt[(1 + x) (1 - 3 x)], {x, 0, 30}], x], 2]] (* _Harvey P. Dale_, May 11 2013 *)
%t Table[2 Sum[Binomial[n, 2 k] Binomial[2 k + 1, k + 1], {k, 0, Floor[n/2]}], {n, 1, 25}] (* _G. C. Greubel_, Mar 01 2017 *)
%t Rest[Rest[CoefficientList[Series[((1 - x)^2 - (1 - x) Sqrt[1 - 2 x - 3 x^2])/(x Sqrt[1 - 2 x - 3 x^2]), {x, 0, 15}], x]]] (* _G. C. Greubel_, Mar 02 2017 *)
%o (PARI) x='x +O('x^50); Vec(((1-x)^2-(1-x +2*x^2)*sqrt(1-2*x-3*x^2)) /(x*sqrt(1 - 2*x -3*x^2))) \\ _G. C. Greubel_, Mar 01 2017
%Y Cf. A025179.
%K nonn
%O 3,1
%A _Clark Kimberling_
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