A158495 Expansion of ((1-4*x) + sqrt(1-4*x))/(2*(1-2*x)).

1, -1, -3, -8, -21, -56, -154, -440, -1309, -4048, -12958, -42712, -144210, -496432, -1735764, -6145968, -21986781, -79331232, -288307254, -1054253208, -3875769606, -14315659632, -53097586284, -197677736208, -738415086066

Offset

0,3

Comments

Hankel transform is A158496.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

a(n) = (2*0^n - 2^n + A126966(n))/2.

Conjecture: n*a(n) +6*(-n+1)*a(n-1) +4*(2*n-3)*a(n-2)=0. - R. J. Mathar, Dec 03 2014

From G. C. Greubel, Jan 09 2023: (Start)

a(n) = [n=0] - Sum_{k=1..n} 2^(n-k)*A000108(k-1).

a(n) = Sum_{j=0..n} 2^(n-j)*A246432(j). (End)

Mathematica

CoefficientList[Series[((1-4x)+Sqrt[1-4x])/(2(1-2x)), {x, 0, 30}], x] (* Harvey P. Dale, Dec 15 2011 *)

Prog

(Magma)
A158495:= func< n | n eq 0 select 1 else - (&+[2^(n-j)*Catalan(j-1): j in [1..n]]) >;
[A158495(n): n in [0..40]]; // G. C. Greubel, Jan 09 2023
(SageMath)
def A158495(n): return int(n==0) - sum(2^(n-k)*catalan_number(k-1) for k in range(1, n+1))
[A158495(n) for n in range(41)] # G. C. Greubel, Jan 09 2023

Crossrefs

Essentially the same as A014318, up to sign and offset.

Cf. A126966, A158496, A246432.

Sequence in context: A008909 A006835 A014318 * A349186 A273720 A018037

Adjacent sequences: A158492 A158493 A158494 * A158496 A158497 A158498

Keyword

easy,sign

Author

Paul Barry, Mar 20 2009

Status

approved