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G.f. A(x) satisfies A(x) = ( 1 + 4*x*(1 + x)*A(x) )^(1/2).
1

%I #9 Apr 17 2024 08:33:53

%S 1,2,4,4,0,-8,-8,16,48,0,-192,-256,480,1792,320,-7808,-12544,19200,

%T 87808,33280,-382464,-720896,854016,4870144,2844160,-20553728,

%U -44909568,38967296,289456128,228986880,-1163309056,-2932514816,1701978112,17951686656,17979965440

%N G.f. A(x) satisfies A(x) = ( 1 + 4*x*(1 + x)*A(x) )^(1/2).

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

%F a(n) = Sum_{k=0..n} 4^k * binomial(k/2+1/2,k) * binomial(k,n-k)/(k+1).

%o (PARI) my(N=40, x='x+O('x^N)); Vec(1/(-2*x*(1+x)+sqrt(1+(2*x*(1+x))^2)))

%o (PARI) a(n) = sum(k=0, n, 4^k*binomial(k/2+1/2, k)*binomial(k, n-k)/(k+1));

%Y Cf. A372035.

%K sign

%O 0,2

%A _Seiichi Manyama_, Apr 16 2024