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A099323 Expansion of (sqrt(1+3*x) + sqrt(1-x))/(2*sqrt(1-x)). 9
1, 1, 0, 1, -1, 3, -6, 15, -36, 91, -232, 603, -1585, 4213, -11298, 30537, -83097, 227475, -625992, 1730787, -4805595, 13393689, -37458330, 105089229, -295673994, 834086421, -2358641376, 6684761125, -18985057351, 54022715451, -154000562758, 439742222071, -1257643249140 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Binomial transform is A072100. Signed Motzkin numbers with an additional leading 1.

Inverse binomial transform of A001405 gives this without the initial 1. So does the binomial transform of (-1)^n*A000108(n) = [1,-1,2,-5,14,-42,...]. - Philippe Deléham, Mar 20 2007

LINKS

Table of n, a(n) for n=0..32.

C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC'02 Melbourne, 2002.

FORMULA

a(n) = 0^n + Sum_{k=0..n-1} binomial(n-1,k)*(-1)^k*C(k), where C(k) is the k-th Catalan number.

G.f.: 1 + x/(1-sqrt(x))/G(0), where G(k)= 1 +  sqrt(x)/(1  -  sqrt(x)/(1 + x/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 28 2013

Conjecture: n*a(n) + 2*(n-2)*a(n-1) + 3*(-n+2)*a(n-2) = 0. - R. J. Mathar, Oct 10 2014

a(n) ~ -(-1)^n * 3^(n + 1/2) / (8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 31 2017

MAPLE

with(PolynomialTools): CoefficientList(convert(taylor((sqrt(1 + 3*x) + sqrt(1 - x))/2/sqrt(1 - x), x = 0, 33), polynom), x); # Taras Goy, Aug 07 2017

MATHEMATICA

CoefficientList[Series[(Sqrt[1+3x]+Sqrt[1-x])/(2Sqrt[1-x]), {x, 0, 40}], x] (* Harvey P. Dale, Feb 06 2015 *)

CROSSREFS

Cf. A000108, A005043.

Sequence in context: A033192 A174297 A005043 * A058534 A063778 A279374

Adjacent sequences:  A099320 A099321 A099322 * A099324 A099325 A099326

KEYWORD

easy,sign

AUTHOR

Paul Barry, Oct 12 2004

EXTENSIONS

Edited by N. J. A. Sloane, Oct 05 2009

STATUS

approved

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Last modified February 24 05:48 EST 2018. Contains 299597 sequences. (Running on oeis4.)