A064331 Generalized Catalan numbers C(-9; n).

1, 1, -8, 145, -3266, 82342, -2223818, 62912809, -1840413050, 55217088622, -1689752866904, 52538652432586, -1655036407913948, 52708355827445800, -1694246075896308110, 54894923324331676345, -1790984858946499478330

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..640

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (19 +Sqrt(1+36*x))/(2*(10-x)) )); // G. C. Greubel, May 03 2019

Mathematica

CoefficientList[Series[(19 +Sqrt[1+36*x])/(2*(10-x)), {x, 0, 30}], x] (* G. C. Greubel, May 03 2019 *)

PARI

my(x='x+O('x^30)); Vec((19 +sqrt(1+36*x))/(2*(10-x))) \\ G. C. Greubel, May 03 2019

Sage

((19 +sqrt(1+36*x))/(2*(10-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)*(-9)^m/n.

a(n) = (1/10)^n*(1 + 9*Sum_{k=0..n-1} C(k)*(-9*10)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).

G.f.: (1+9*x*c(-9*x)/10)/(1-x/10) = 1/(1-x*c(-9*x)) with c(x) g.f. of Catalan numbers A000108.