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%I #20 Jul 22 2022 11:42:55
%S 0,0,1,3,9,20,50,105,245,504,1134,2310,5082,10296,22308,45045,96525,
%T 194480,413270,831402,1755182,3527160,7407036,14872858,31097794,
%U 62403600,130007500,260757900,541574100,1085822640,2249204040,4508102925,9316746045
%N Number of turns in all left factors of Dyck paths of length n.
%H G. C. Greubel, <a href="/A191527/b191527.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{k=0..n} k*binomial(floor((n-1)/2), floor(k/2))*binomial(ceiling((n-1)/2), ceiling(k/2)).
%F G.f.: g(z)=2*z^2*(1-4*z^2-4*z^3)/((1-2*z)*((1+z)*(1-4*z^2)*(1-2*z)+(1-z-4*z^2)*sqrt(1-4*z^2))).
%F a(n) ~ 2^(n-1/2)*sqrt(n)/sqrt(Pi). - _Vaclav Kotesovec_, Mar 21 2014
%F D-finite with recurrence (n+1)*a(n) + (-n-1)*a(n-1) + 2*(-4*n+5)*a(n-2) + 4*(n+1)*a(n-3) + 16*(n-3)*a(n-4) = 0. - _R. J. Mathar_, Jun 06 2014
%e a(4)=9 because in UDUD, UDUU, UUDD, UUDU, UUUD, and UUUU we have a total of 3+2+1+2+1+0=9 turns (here U=(1,1) and D=(1,-1)).
%p g := 2*z^2*(1-4*z^2-4*z^3)/((1-2*z)*((1+z)*(1-4*z^2)*(1-2*z)+(1-z-4*z^2)*sqrt(1-4*z^2))): gser := series(g, z = 0, 35): seq(coeff(gser, z, n), n = 0 .. 32);
%p a := proc (n) options operator, arrow: sum(k*binomial(floor((1/2)*n-1/2), floor((1/2)*k))*binomial(ceil((1/2)*n-1/2), ceil((1/2)*k)), k = 0 .. n) end proc: seq(a(n), n = 0 .. 32);
%t CoefficientList[Series[2*x^2*(1-4*x^2-4*x^3)/((1-2*x)*((1+x)*(1-4*x^2)*(1-2*x)+(1-x-4*x^2)*Sqrt[1-4*x^2])), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 21 2014 *)
%o (PARI) x='x+O('x^50); concat([0,0], Vec(2*x^2*(1-4*x^2-4*x^3)/((1-2*x)*((1+x)*(1-4*x^2)*(1-2*x)+(1-x-4*x^2)*sqrt(1-4*x^2))))) \\ _G. C. Greubel_, May 27 2017
%Y Cf. A088855.
%K nonn
%O 0,4
%A _Emeric Deutsch_, Jun 06 2011
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