A064328 Generalized Catalan numbers C(-6; n).

1, 1, -5, 61, -917, 15421, -277733, 5239117, -102188021, 2044131037, -41706059525, 864547613293, -18157111255829, 385517710342909, -8261602828082213, 178459989617336461, -3881680470161846837

Offset: 0

Author

Wolfdieter Lang, Sep 21 2001

Keywords

• sign
• easy

Comments

See triangle A064334 with columns m built from C(-m; n), m >= 0, also for Derrida et al. references.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..720

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (13 +Sqrt(1+24*x))/(2*(7-x)) )); // G. C. Greubel, May 03 2019

Mathematica

CoefficientList[Series[(13 +Sqrt[1+24*x])/(2*(7-x)), {x, 0, 30}], x] (* G. C. Greubel, May 03 2019 *)

PARI

my(x='x+O('x^30)); Vec((13 +sqrt(1+24*x))/(2*(7-x))) \\ G. C. Greubel, May 03 2019

Sage

((13 +sqrt(1+24*x))/(2*(7-x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 03 2019

Formula

a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(-6)^m/n.

a(n) = (1/7)^n*(1 + 6*Sum_{k=0..n-1} C(k)*(-6*7)^k), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).

G.f.: (1+6*x*c(-6*x)/7)/(1-x/7) = 1/(1-x*c(-6*x)) with c(x) g.f. of Catalan numbers A000108.