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A099323
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Expansion of (sqrt(1+3*x) + sqrt(1-x))/(2*sqrt(1-x)).
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16
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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
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OFFSET
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0,6
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COMMENTS
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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
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LINKS
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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CoefficientList[Series[(Sqrt[1+3x]+Sqrt[1-x])/(2Sqrt[1-x]), {x, 0, 40}], x] (* Harvey P. Dale, Feb 06 2015 *)
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PROG
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(Magma)
A099323:= func< n | (&+[(-1)^k*Binomial(n-1, k)*Catalan(k): k in [0..n]]) >;
(Sage) [sum((-1)^k*binomial(n-1, k)*catalan_number(k) for k in (0..n)) for n in (0..40)] # G. C. Greubel, Nov 25 2021
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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