A064329 Generalized Catalan numbers C(-7; n).

1, 1, -6, 85, -1490, 29226, -614004, 13511709, -307448490, 7174776190, -170777485556, 4130050311234, -101192982385844, 2506610481299380, -62668163792277840, 1579300030107459885, -40076101342241993370

Offset

0,3

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..690

Formula

a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(-7)^m/n.

a(n) = (1/8)^n*(1 + 7*Sum_{k=0..n-1} C(k)*(-7*8)^k), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).

G.f.: (1+7*x*c(-7*x)/8)/(1-x/8) = 1/(1-x*c(-7*x)) with c(x) g.f. of Catalan numbers A000108.

Mathematica

CoefficientList[Series[(15 +Sqrt[1+28*x])/(2*(8-x)), {x, 0, 30}], x] (* G. C. Greubel, May 03 2019 *)

Prog

(PARI) my(x='x+O('x^30)); Vec((15 +sqrt(1+28*x))/(2*(8-x))) \\ G. C. Greubel, May 03 2019
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (15 +Sqrt(1+28*x))/(2*(8-x)) )); // G. C. Greubel, May 03 2019
(Sage) ((15 +sqrt(1+28*x))/(2*(8-x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 03 2019

Crossrefs

Sequence in context: A164266 A136597 A381243 * A187740 A332407 A358297

Adjacent sequences: A064326 A064327 A064328 * A064330 A064331 A064332

Keyword

sign,easy

Author

Wolfdieter Lang, Sep 21 2001

Status

approved