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A099323 Expansion of (sqrt(1+3*x) + sqrt(1-x))/(2*sqrt(1-x)). 16
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 *)
PROG
(Magma)
A099323:= func< n | (&+[(-1)^k*Binomial(n-1, k)*Catalan(k): k in [0..n]]) >;
[A099323(n): n in [0..40]]; // G. C. Greubel, Nov 25 2021
(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
CROSSREFS
Sequence in context: A033192 A174297 A005043 * A342912 A370241 A058534
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 March 19 04:26 EDT 2024. Contains 370952 sequences. (Running on oeis4.)