A099323 Expansion of (sqrt(1+3*x) + sqrt(1-x))/(2*sqrt(1-x)).

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

Offset: 0

Author

Paul Barry, Oct 12 2004

Keywords

• easy
• sign

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

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC'02 Melbourne, 2002.

Crossrefs

Cf. A000108, A001405, A005043, A072100.

Programs

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

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 *)

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

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

Extensions

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