A156857 Expansion of (1+2*x)/(1+x+4*x^2)^2.

1, 0, -9, 10, 45, -108, -125, 702, -135, -3320, 4239, 11250, -31931, -18180, 165915, -92762, -651375, 1101168, 1747495, -6710310, -694179, 30182500, -28394829, -101934450, 229069225, 203510232, -1198850625, 364506562, 4767453045

Offset: 0

Author

Paul Barry, Feb 17 2009

Keywords

• easy
• sign

Comments

Hankel transform of A091526.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-2,-9,-8,-16).

Crossrefs

Cf. A091526, A106853.

Programs

Magma

I:=[1,0,-9,10]; [n le 4 select I[n] else (-1)*(2*Self(n-1) +9*Self(n-2) +8*Self(n-3) +16*Self(n-4)): n in [1..41]]; // G. C. Greubel, Jan 28 2022

Mathematica

LinearRecurrence[{-2,-9,-8,-16}, {1,0,-9,10}, 41] (* or *)
A156857[n_]:= (-2)^n*Sum[ChebyshevU[n-j, 1/4]*(ChebyshevU[j, 1/4] - ChebyshevU[j-1, 1/4]), {j,0,n}];
Table[A156857[n], {n, 0, 40}] (* G. C. Greubel, Jan 28 2022 *)

Sage

def A156857(n): return (-2)^n*sum( chebyshev_U(n-j, 1/4)*(chebyshev_U(j, 1/4) - chebyshev_U(j-1, 1/4)) for j in (0..n))
[A156857(n) for n in (0..40)] # G. C. Greubel, Jan 28 2022

Formula

G.f.: (1 +2*x)/(1 +2*x +9*x^2 +8*x^3 +16*x^4).

a(n) = (-2)*Sum_{j=0..n} ChebyshevU(n-j, 1/4)*(ChebyshevU(j, 1/4) - ChebyshevU(j-1, 1/4)). - G. C. Greubel, Jan 28 2022