A115145 Sixth convolution of A115140.

1, -6, 9, -2, 0, 0, -1, -6, -27, -110, -429, -1638, -6188, -23256, -87210, -326876, -1225785, -4601610, -17298645, -65132550, -245642760, -927983760, -3511574910, -13309856820, -50528160150, -192113383644, -731508653106, -2789279908316, -10649977831752, -40715807302800

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1669

Formula

O.g.f.: 1/c(x)^6 = P(7, x) - x*P(6, 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(7, x) = 1-5*x+6*x^2-x^3 and P(6, x) = 1-4*x+3*x^2.

a(n) = -C6(n-6), n>=6, with C6(n) = A003517(n+2) (sixth convolution of Catalan numbers). a(0) = 1, a(1) = -6, a(2) = 9, a(3) = -2, a(4) = a(5) = 0. [1, -6, 9, -2] is row n = 6 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.

D-finite with recurrence n*(n-6)*a(n) - 2*(2*n-7)*(n-4)*a(n-1) = 0. - R. J. Mathar, Sep 23 2021

From Amiram Eldar, Oct 09 2025: (Start)

a(n) = -6 * binomial(2*n-7, n-6)/n for n >= 1.

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

Mathematica

CoefficientList[Series[(1-6*x+9*x^2-2*x^3 +(1-4*x+3*x^2)*Sqrt[1-4*x])/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 12 2019 *)

Prog

(PARI) my(x='x+O('x^30)); Vec((1-6*x+9*x^2-2*x^3 +(1-4*x+3*x^2) *sqrt(1-4*x))/2) \\ G. C. Greubel, Feb 12 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-6*x+9*x^2-2*x^3 +(1-4*x+3*x^2)*Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 12 2019
(SageMath) ((1-6*x+9*x^2-2*x^3 +(1-4*x+3*x^2)*sqrt(1-4*x))/2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019

Crossrefs

Cf. A000108, A003517, A034807.

Cf. A115139, A115140, A115141, A115142, A115143.

Cf. A115144, A115146, A115147, A115148, A115149.

Sequence in context: A011454 A379105 A274480 * A296478 A195403 A021595

Adjacent sequences: A115142 A115143 A115144 * A115146 A115147 A115148

Keyword

sign,easy

Author

Wolfdieter Lang, Jan 13 2006

Status

approved