A064332 Generalized Catalan numbers C(-10; n).

1, 1, -9, 181, -4529, 126861, -3806649, 119653941, -3889122369, 129646443421, -4408213959689, 152290162367301, -5330337257966609, 188617242067457581, -6736489341630231129, 242518500968942706261, -8791448318093732481249

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..625

Crossrefs

Cf. A000108, A064334.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (21 +Sqrt(1+40*x))/(2*(11-x)) )); // G. C. Greubel, May 03 2019

Mathematica

CoefficientList[Series[(21 +Sqrt[1+40*x])/(2*(11-x)), {x, 0, 30}], x] (* G. C. Greubel, May 03 2019 *)

PARI

my(x='x+O('x^30)); Vec((21 +sqrt(1+40*x))/(2*(11-x))) \\ G. C. Greubel, May 03 2019

Sage

((21 +sqrt(1+40*x))/(2*(11-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)*(-10)^m/n.

a(n) = (1/11)^n*(1 + 10*Sum_{k=0..n-1} C(k)*(-10*11)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).

G.f.: (1+10*x*c(-10*x)/11)/(1-x/11) = 1/(1-x*c(-10*x)) with c(x) g.f. of Catalan numbers A000108.