A378816 Expansion of 2*(x - 1)^3/(3*x^3 - 5*x^2 + x + 1 + sqrt(-(x - 1)^3*(x + 1)^2*(3*x + 1))).

-1, 4, -11, 30, -83, 232, -654, 1856, -5296, 15180, -43675, 126062, -364863, 1058552, -3077533, 8963862, -26151753, 76409052, -223544241, 654790218, -1920055017, 5635816776, -16557539124, 48685404516, -143264248974, 421879104836, -1243160223829, 3665516301186

Offset: 0

Author

Thomas Scheuerle, Dec 08 2024

Keywords

• sign

Comments

Binomial transform of A057552(n)*(-1)^(n+1).

Crossrefs

Cf. A025566, A057552, A378783, A378816 ( Hankel sequence transform ).

Programs

PARI

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

Formula

G.f. A(x) satisfies: (-3*x^3 - x^2)*A(x)^2 + (3*x^3 - 5*x^2 + x + 1)*A(x) + (-x^3 + x*y^2 - x*y + 1) = 0.

a(n) = Limit_{k->oo} (A378783(k, k-n) - A378783(k, k-n-1)).

a(n) = A025566(n+1)+A025566(n+2)*(-1)^(n+1), for n > 0.

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