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 A099323 Expansion of (sqrt(1+3*x) + sqrt(1-x))/(2*sqrt(1-x)). 15
 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 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 D-finite with recurrence: 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 * A342912 A058534 A063778 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 April 23 12:15 EDT 2021. Contains 343204 sequences. (Running on oeis4.)