A372023 Expansion of ( (1 + 3*x)/(1 - x) )^(1/2).

1, 2, 0, 2, -2, 6, -12, 30, -72, 182, -464, 1206, -3170, 8426, -22596, 61074, -166194, 454950, -1251984, 3461574, -9611190, 26787378, -74916660, 210178458, -591347988, 1668172842, -4717282752, 13369522250, -37970114702, 108045430902, -308001125516

Offset: 0

Author

Seiichi Manyama, Apr 16 2024

Keywords

• sign

Crossrefs

Cf. A005043, A103872, A372024.

Programs

PARI

my(N=40, x='x+O('x^N)); Vec(sqrt((1+3*x)/(1-x)))

PARI

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

Formula

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

a(n) = (-1)^(n-1) * 2 * A005043(n-1) for n > 0.

a(n) ~ (-1)^(n+1) * 3^(n + 1/2) / (4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 16 2024