A064330 Generalized Catalan numbers C(-8; n).

1, 1, -7, 113, -2263, 50721, -1217703, 30622929, -796311415, 21237226625, -577699502407, 15966537989425, -447086291268119, 12656524451911393, -361628025405250023, 10415207118205622673, -302049007052246016183

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..660

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (17 +Sqrt(1+32*x))/(2*(9-x)) )); // G. C. Greubel, May 03 2019

Mathematica

CoefficientList[Series[(17 +Sqrt[1+32*x])/(2*(9-x)), {x, 0, 30}], x] (* G. C. Greubel, May 03 2019 *)

PARI

my(x='x+O('x^30)); Vec((17 +sqrt(1+32*x))/(2*(9-x))) \\ G. C. Greubel, May 03 2019

Sage

((17 +sqrt(1+32*x))/(2*(9-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)*(-8)^m/n.

a(n) = (1/9)^n*(1 + 8*Sum_{k=0..n-1} C(k)*(-8*9)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).

G.f.: (1+8*x*c(-8*x)/9)/(1-x/9) = 1/(1-x*c(-8*x)) with c(x) g.f. of Catalan numbers A000108.