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%I #24 Oct 23 2023 09:49:34
%S 1,3,10,34,117,407,1430,5070,18122,65246,236436,861764,3157325,
%T 11622015,42961470,159419670,593636670,2217608250,8308432140,
%U 31212003420,117544456770,443690433654,1678353186780,6361322162444,24155384502452,91882005146652
%N a(n) = binomial(2*n-1, n) - binomial(2*n-1, n+3).
%C Number of (s(0), s(1), ..., s(2n-1)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 2, s(2n-1) = 3. Also a(n) = T(2n-1,n-1), where T is the array defined in A026009.
%C Number of integer lattice paths from (0,2) to (n-1,n+2) that do not cross the main diagonal.
%H Vincenzo Librandi, <a href="/A026016/b026016.txt">Table of n, a(n) for n = 1..200</a>
%F Expansion of (1+x^1*C^3)*C^2, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
%F (n+3)*a(n) +(-7*n-9)*a(n-1) +2*(7*n-4)*a(n-2) +4*(-2*n+5)*a(n-3)=0. - _R. J. Mathar_, Jun 20 2013
%F From _G. C. Greubel_, Mar 19 2021: (Start)
%F G.f.: (1-x)*( 1 -4*x +2*x^2 - (1-2*x)*sqrt(1-4*x) )/(2*x^3).
%F E.g.f.: -1 + (exp(2*x)/x^2)*( x*(1+x)*BesselI(0, 2*x) - (2 +x +2*x^2)*Bessel(1, 2*x) ).
%F a(n) = C(n) + Sum_{j=0..n-2} C(n-j-2)*(C(j+3) - 2*C(j+2)), where C(n) are the Catalan numbers. (End)
%F a(n) = C(n+2) -3*C(n+1) +2*C(n) = 6*((n^2+1)/((n+2)*(n+3)))*C(n). - _G. C. Greubel_, Mar 22 2021
%p a:= n-> binomial(2*n-1, n) -binomial(2*n-1, n+3): seq(a(n), n=1..27); # _Zerinvary Lajos_, Dec 10 2007
%t Table[Binomial[2 n - 1, n] - Binomial[2 n - 1, n + 3], {n, 1, 40}] (* _Vincenzo Librandi_, Jun 21 2013 *)
%o (Magma) [Binomial(2*n-1, n) - Binomial(2*n-1, n+3): n in [1..30]]; // _Vincenzo Librandi_, Jun 21 2013
%o (Sage) [binomial(2*n-1, n) - binomial(2*n-1, n+3) for n in (2..30)] # _G. C. Greubel_, Mar 19 2021
%o (PARI) a(n) = binomial(2*n-1, n) - binomial(2*n-1, n+3) \\ _Charles R Greathouse IV_, Oct 23 2023
%Y Cf. A000108, A026009.
%K nonn
%O 1,2
%A _Clark Kimberling_
%E Better description from Darko Marinov (marinov(AT)lcs.mit.edu), May 17 2001
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