A115146 Seventh convolution of A115140.

1, -7, 14, -7, 0, 0, 0, -1, -7, -35, -154, -637, -2548, -9996, -38760, -149226, -572033, -2187185, -8351070, -31865925, -121580760, -463991880, -1771605360, -6768687870, -25880277150, -99035193894, -379300783092, -1453986335186, -5578559816632, -21422369201800

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1670

Formula

O.g.f.: 1/c(x)^7 = P(8, x) - x*P(7, 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(8, x) = 1-6*x+10*x^2-4*x^3 and P(7, x) = 1-5*x+6*x^2-x^3.

a(n) = -C7(n-7), n>=7, with C7(n) = A000588(n+3) (seventh convolution of Catalan numbers). a(0) = 1, a(1) = -7, a(2) = 14, a(3) = -7, a(4) = a(5) = a(6) = 0. [1, -7, 14, -7] is row n = 7 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.

D-finite with recurrence n*(n-7)*a(n) - 2*(n-4)*(2*n-9)*a(n-1) = 0. - R. J. Mathar, Sep 15 2024

From Amiram Eldar, Oct 09 2025: (Start)

a(n) = -7 * binomial(2*n-8, n-7)/n for n >= 1.

a(n) ~ -7 * 4^(n-4) / (n^(3/2) * sqrt(Pi)). (End)

Mathematica

CoefficientList[Series[(1-7*x+14*x^2-7*x^3 +(1-5*x+6*x^2-x^3) *Sqrt[1-4*x])/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 12 2019 *)

Prog

(PARI) my(x='x+O('x^30)); Vec((1-7*x+14*x^2-7*x^3 +(1-5*x+6*x^2-x^3) *sqrt(1-4*x))/2) \\ G. C. Greubel, Feb 12 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-7*x+14*x^2-7*x^3 +(1-5*x+6*x^2-x^3)*Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 12 2019
(SageMath) ((1-7*x+14*x^2-7*x^3 +(1-5*x+6*x^2-x^3)*sqrt(1-4*x))/2 ).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019

Crossrefs

Cf. A000108, A000588, A034807.

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

Cf. A115144, A115145, A115147, A115148, A115149.

Sequence in context: A258767 A169845 A302972 * A340312 A029844 A000730

Adjacent sequences: A115143 A115144 A115145 * A115147 A115148 A115149

Keyword

sign,easy

Author

Wolfdieter Lang, Jan 13 2006

Status

approved