A372002 G.f. A(x) satisfies A(x) = ( 1 + 4*x*(1 + x*A(x)) )^(1/2).

1, 2, 0, 4, -8, 24, -72, 224, -720, 2368, -7936, 27008, -93088, 324288, -1140032, 4039296, -14409728, 51713792, -186577152, 676334592, -2462090752, 8997154816, -32992079872, 121362092032, -447721572864, 1656081763328, -6140640246784, 22820403312640

Offset: 0

Author

Seiichi Manyama, Apr 15 2024

Keywords

• sign

Crossrefs

Cf. A372003.

Programs

Maple

A372002 := proc(n)
    add(4^k*binomial((n-k+1)/2,k)*binomial(k,n-k)/(n-k+1),k=0..n) ;
end proc:
seq(A372002(n),n=0..60) ; # R. J. Mathar, Apr 22 2024

PARI

my(N=30, x='x+O('x^N)); Vec((1+4*x)/(-2*x^2+sqrt(1+4*x+4*x^4)))

PARI

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

Formula

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

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

D-finite with recurrence n*a(n) +2*(2*n-3)*a(n-1) +4*(n-6)*a(n-4)=0. - R. J. Mathar, Apr 22 2024