A292440 -id:A292440

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A292461

Expansion of (1 - x - x^2 + sqrt((1 - x - x^2)^2 - 4*x^3))/2 in powers of x.

+10
2

1, -1, -1, -1, -1, -2, -4, -8, -17, -37, -82, -185, -423, -978, -2283, -5373, -12735, -30372, -72832, -175502, -424748, -1032004, -2516347, -6155441, -15101701, -37150472, -91618049, -226460893, -560954047, -1392251012, -3461824644, -8622571758, -21511212261

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

FORMULA

Convolution inverse of A292460.

Let f(x) = (1 - x - x^2 - sqrt((1 - x - x^2)^2 - 4*x^3))/(2*x^3).

G.f.: 1-x-x^2-x^3/(1-x-x^2-x^3/(1-x-x^2-x^3/(1-x-x^2-x^3/(1-x-x^2-x^3/(... (continued fraction).

G.f.: 1/f(x) = 1 - x - x^2 - x^3*f(x).

a(n) = -A292460(n-3) for n > 2.

a(n) ~ -5^(1/4) * phi^(2*n - 2) / (2 * sqrt(Pi) * n^(3/2)), where phi is the golden ratio (1+sqrt(5))/2. - Vaclav Kotesovec, Sep 17 2017, simplified Dec 06 2021

Conjecture D-finite with recurrence: n*a(n) +(-2*n+3)*a(n-1) +(-n+3)*a(n-2) +(-2*n+9)*a(n-3) +(n-6)*a(n-4)=0. - R. J. Mathar, Jan 24 2020

MATHEMATICA

CoefficientList[Series[(1-x-x^2 +Sqrt[(1-x-x^2)^2 -4*x^3])/2, {x, 0, 50} ], x] (* G. C. Greubel, Aug 13 2018 *)

PROG

(PARI) x='x+O('x^50); Vec((1-x-x^2 +sqrt((1-x-x^2)^2 -4*x^3))/2) \\ G. C. Greubel, Aug 13 2018

(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x-x^2 +Sqrt((1-x-x^2)^2 -4*x^3))/2)); // G. C. Greubel, Aug 13 2018

CROSSREFS

Cf. A004148, A203019, A292440, A292460.

KEYWORD

sign

AUTHOR

Seiichi Manyama, Sep 16 2017

STATUS

approved

A308846

Expansion of e.g.f. x*exp(-x) / BesselI(1,2*x).

+10
0

1, -1, 0, 2, -1, -11, 11, 125, -181, -2443, 4534, 73116, -164075, -3106169, 8150624, 177689590, -533231545, -13167063415, 44461467464, 1226832808294, -4603245727023, -140381127928757, 579412786145335, 19352381237997941, -87137414891475575, -3163451350469601721, 15431005593881754746

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

E.g.f. is inverse of e.g.f. for A001006 (Motzkin numbers).

LINKS

Table of n, a(n) for n=0..26.

FORMULA

E.g.f.: 1 / Sum_{k>=0} A001006(k)*x^k/k!.

MATHEMATICA

nmax = 26; CoefficientList[Series[x Exp[-x]/BesselI[1, 2 x], {x, 0, nmax}], x] Range[0, nmax]!

a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n, k] Hypergeometric2F1[(1 - k)/2, -k/2, 2, 4] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 26}]

PROG

(PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(-x) / besseli(1, 2*x))) \\ Michel Marcus, Jul 02 2019