A115148 Ninth convolution of A115140.

1, -9, 27, -30, 9, 0, 0, 0, 0, -1, -9, -54, -273, -1260, -5508, -23256, -95931, -389367, -1562275, -6216210, -24582285, -96768360, -379629720, -1485507600, -5801732460, -22626756594, -88152205554, -343176898988, -1335293573130, -5193831553416, -20198233818840

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1671

Formula

O.g.f.: 1/c(x)^9 = P(10, x) - x*P(9, x)*c(x) with the o.g.f. c(x) = (1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers) and the polynomials P(n, x) defined in A115139. Here P(10, x) = 1-8*x+21*x^2-20*x^3+5*x^4 and P(9, x) = 1-7*x+15*x^2-10*x^3+x^4.

a(n) = -C9(n-9), n>=9, with C9(n) = A001392(n+4) (eighth convolution of Catalan numbers). a(0) = 1, a(1) = -9, a(2) = 27, a(3) = -30, a(4) = 9, a(5) = a(6) = a(7) = a(8) = 0. [1, -9, 27, -30, 9] is row n=9 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.

From Amiram Eldar, Oct 09 2025: (Start)

a(n) = -9 * binomial(2*n-10, n-9)/n for n >= 1.

a(n) ~ -9 * 4^(n-5) / (n^(3/2) * sqrt(Pi)). (End)

Mathematica

CoefficientList[Series[(1-9*x+27*x^2-30*x^3+9*x^4 +(1-7*x+15*x^2-10*x^3 +x^4)*Sqrt[1-4*x])/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 12 2019 *)

Prog

(PARI) my(x='x+O('x^30)); Vec((1-9*x+27*x^2-30*x^3+9*x^4 +(1-7*x+15*x^2 -10*x^3+x^4)*sqrt(1-4*x))/2) \\ G. C. Greubel, Feb 12 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-9*x+27*x^2-30*x^3+9*x^4 +(1-7*x+15*x^2-10*x^3+x^4)*Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 12 2019
(SageMath) ((1-9*x+27*x^2-30*x^3+9*x^4 +(1-7*x+15*x^2-10*x^3+x^4) *sqrt(1-4*x))/2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019

Crossrefs

Cf. A000108, A001392, A034807, A115139 - A115147, A115149.

Sequence in context: A255622 A209511 A370872 * A022701 A276003 A255343

Adjacent sequences: A115145 A115146 A115147 * A115149 A115150 A115151

Keyword

sign,easy

Author

Wolfdieter Lang, Jan 13 2006

Status

approved