A064333 Generalized Catalan numbers C(-11; n).

1, 1, -10, 221, -6082, 187386, -6184848, 213843477, -7645509706, 280351640702, -10485617230780, 398467433529298, -15341431926699284, 597149747213056324, -23459916801814723548, 929028306450848244741, -37045540042729366580442

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..600

Crossrefs

Cf. A000108, A064334.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (23 +Sqrt(1+44*x))/(2*(12-x)) )); // G. C. Greubel, May 03 2019

Mathematica

CoefficientList[Series[(23 +Sqrt[1+44*x])/(2*(12-x)), {x, 0, 30}], x] (* G. C. Greubel, May 03 2019 *)

PARI

my(x='x+O('x^30)); Vec((23 +sqrt(1+44*x))/(2*(12-x))) \\ G. C. Greubel, May 03 2019

Sage

((23 +sqrt(1+44*x))/(2*(12-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)*(-11)^m/n.

a(n) = (1/12)^n*(1 + 11*Sum_{k=0..n-1} C(k)*(-11*12)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).

G.f.: (1+11*x*c(-11*x)/12)/(1-x/12) = 1/(1-x*c(-11*x)) with c(x) g.f. of Catalan numbers A000108.