%I #51 Feb 22 2024 09:38:09
%S -1,-1,0,7,42,198,858,3575,14586,58786,235144,936054,3714500,14709420,
%T 58169070,229824855,907513530,3582290250,14138105520,55795023570,
%U 220196403180,869084354580,3430596136500,13543993546902,53481410415972,211224560329748,834402610992048
%N Expansion of (3*x/2 - 1 - (7*x - 2)/(2*sqrt(1 - 4*x)))/x.
%H G. C. Greubel, <a href="/A246434/b246434.txt">Table of n, a(n) for n = 1..1000</a>
%F G.f.: x*(3*x*C(x)-1)/((1-2*x*C(x))*(1-x*C(x))^2), where C(x) is the g.f. of A000108.
%F a(n) = Sum_{k=1..n} k*(-1)^k*binomial(n-1, k-1)*binomial(3*n-k-1, n-k))/n.
%F a(n) = (2^(2*n-1)*(n-3)*(n-1/2)!)/(sqrt(Pi)*(n+1)!). - _Peter Luschny_, Nov 14 2014
%F D-finite with recurrence (for n > 4): (n-4)*(n+1)*a(n) = 2*(n-3)*(2*n-1)*a(n-1). - _Vaclav Kotesovec_, Nov 14 2014
%F G.f.: x^4* 2F1(2,9/2;6;4*x) -x -x^2. - _R. J. Mathar_, Jan 25 2020
%F From _Peter Bala_, Feb 13 2024: (Start)
%F a(n) = 2*binomial(2*n-1, n+1) - binomial(2*n-1, n) = binomial(2*n-1, n+1) - Catalan(n).
%F a(n) is odd iff n = 2^k for k >= 0. (End)
%t Table[Sum[k*(-1)^k*Binomial[n-1,k-1]*Binomial[3*n-k-1,n-k],{k,1,n}]/n,{n,1,20}] (* _Vaclav Kotesovec_, Nov 14 2014 after _Vladimir Kruchinin_ *)
%t Rest[Rest[CoefficientList[Series[1-(7*x-2)/(2*Sqrt[1-4*x]),{x,0,30}],x]]] (* _Vaclav Kotesovec_, Nov 14 2014 *)
%o (Maxima)
%o a(n):=sum(k*(-1)^k*binomial(n-1,k-1)*binomial(3*n-k-1,n-k),k,1,n)/n;
%o (Sage)
%o a = lambda n: (2^(2*n-1)*(n-3)*factorial(n-1/2))/(sqrt(pi)* factorial(n+1))
%o [a(n) for n in (1..20)] # _Peter Luschny_, Nov 14 2014
%o (PARI) x='x+O('x^50); Vec((3*x/2-1-(7*x-2)/(2*sqrt(1-4*x)))/x) \\ _G. C. Greubel_, Jun 02 2017
%Y Cf. A000108.
%K sign
%O 1,4
%A _Vladimir Kruchinin_, Nov 14 2014
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