A364371 - OEIS

%I #18 Aug 28 2024 02:34:28

%S 1,0,-1,2,-2,-1,9,-20,20,24,-150,327,-293,-599,3097,-6452,4854,15878,

%T -71252,140112,-81328,-437346,1746254,-3214989,1223971,12345295,

%U -44552833,76242173,-11292089,-354175849,1167638037,-1842585992,-233903034,10273377388,-31169512310

%N G.f. satisfies A(x) = (1 + x) * (1 - x*A(x)^2).

%F G.f.: A(x) = 2*(1 + x) / (1 + sqrt(1+4*x*(1 + x)^2)).

%F a(n) = Sum_{k=0..n} (-1)^k * binomial(2*k+1,k) * binomial(2*k+1,n-k) / (2*k+1).

%F D-finite with recurrence (n+1)*a(n) +(5*n-1)*a(n-1) +6*(2*n-3)*a(n-2) +6*(2*n-5)*a(n-3) +2*(2*n-7)*a(n-4)=0. - _R. J. Mathar_, Jul 25 2023

%F From _Peter Bala_, Aug 24 2024: (Start)

%F A(x) = (1 + x)*c(-x*(1+x)^2), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.

%F (1/x) * series_reversion(x/A(x)) = 1 - x^2 + 2*x^3 - 11*x^5 + 28*x^6 + ..., the g.f. of A364375. (End)

%p A364371 := proc(n)

%p     add((-1)^k* binomial(2*k+1,k) * binomial(2*k+1,n-k)/(2*k+1),k=0..n) ;

%p end proc:

%p seq(A364371(n),n=0..70); # _R. J. Mathar_, Jul 25 2023

%o (PARI) a(n) = sum(k=0, n, (-1)^k*binomial(2*k+1, k)*binomial(2*k+1, n-k)/(2*k+1));

%Y Cf. A364372, A364373, A364375.

%Y Cf. A000108, A073157.

%K sign,easy

%O 0,4

%A _Seiichi Manyama_, Jul 20 2023