A167479 Convolution of the Catalan numbers A000108(n) and (-2)^n.

1, -1, 4, -3, 20, 2, 128, 173, 1084, 2694, 11408, 35970, 136072, 470756, 1732928, 6228989, 22899692, 83845406, 309947888, 1147367414, 4269385592, 15927495836, 59627571968, 223804469714, 842295207896, 3177355985660, 12012641100832

Offset: 0

Author

Paul Barry, Nov 04 2009

Keywords

• easy
• sign

Comments

Hankel transform is A079935.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500
• S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017)

Crossrefs

Cf. A000108, A079935.

Programs

Mathematica

CoefficientList[Series[(1 - Sqrt[1 - 4*t])/(2*t*(1 + 2*t)), {t, 0, 50}], t] (* G. C. Greubel, Jun 13 2016 *)

Formula

G.f.: c(x)/(1+2x), c(x) the g.f. of A000108.

a(n) = Sum_{k=0..n} (-2)^(n-k)*A000108(k).

(n+1)*a(n) + 2*(2-n)*a(n-1) + 4*(1-2*n)*a(n-2)=0. - R. J. Mathar, Nov 16 2011 [Proof in Ekhad/Yang, Theorem 26]

a(n) ~ 2^(2*n + 1) / (3 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 08 2018