A158495 -id:A158495 - cp's OEIS frontend

A158496 Expansion of (1-4x+x^2)/(1+x^2)^2.

1, -4, -1, 8, 1, -12, -1, 16, 1, -20, -1, 24, 1, -28, -1, 32, 1, -36, -1, 40, 1, -44, -1, 48, 1, -52, -1, 56, 1, -60, -1, 64, 1, -68, -1, 72, 1, -76, -1, 80, 1, -84, -1, 88, 1, -92, -1, 96, 1, -100, -1, 104, 1, -108, -1, 112, 1, -116, -1, 120, 1, -124, -1, 128, 1, -132, -1

Offset: 0

Paul Barry, Mar 20 2009

Hankel transform of A158495.

Index entries for linear recurrences with constant coefficients, signature (0,-2,0,-1).

Cf. A019425.

CoefficientList[Series[(1-4x+x^2)/(1+x^2)^2,{x,0,70}],x] (* or *) LinearRecurrence[{0,-2,0,-1},{1,-4,-1,8},70] (* Harvey P. Dale, Mar 06 2012 *)

Vec((1-4*x+x^2)/(1+x^2)^2 + O(x^100)) \\ Altug Alkan, Jan 10 2016

a(n) = (n+3/2-(n+1/2)*(-1)^n)*(-1)^C(n+1,2).

a(0)=1, a(1)=-4, a(2)=-1, a(3)=8, a(n) = -2*a(n-2)-a(n-4). - Harvey P. Dale, Mar 06 2012

A158499 Expansion of (1 + sqrt(1-4x))/(2-4x).

Original entry on oeis.org

1, 1, 1, 0, -5, -24, -90, -312, -1053, -3536, -11934, -40664, -140114, -488240, -1719380, -6113200, -21921245, -79200160, -288045110, -1053728920, -3874721030, -14313562480, -53093391980, -197669347600, -738398308850, -2766700765024

Offset: 0

Paul Barry, Mar 20 2009

Hankel transform is A056594 with g.f. 1/(1+x^2).
Row sums of the Riordan array (sqrt(1-4*x)/(1-2*x), x*c(x)^2), c(x) the g.f. of A000108.
The inverse Catalan transform yields A146559. - R. J. Mathar, Mar 20 2009

Matthew House, Table of n, a(n) for n = 0..1669
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.

Cf. A000108, A056594, A146559, A158495.

[n le 2 select 1 else (6*(n-2)*Self(n-1) - 4*(2*n-5)*Self(n-2))/(n-1): n in [1..30]]; // G. C. Greubel, Mar 17 2025

CoefficientList[ Series[(1 + Sqrt[1 - 4x])/(2 - 4x), {x, 0, 26}], x] (* Robert G. Wilson v, Nov 08 2015 *)

my(x='x+O('x^33)); Vec(((1-4*x)+sqrt(1-4*x))/(2*(1-2*x)*sqrt(1-4*x))) \\ Altug Alkan, Nov 08 2015

@CachedFunction
def a(n): # a = A158499
    if n<2: return 1
    else: return (6*(n-1)*a(n-1) - 4*(2*n-3)*a(n-2))/n
[a(n) for n in range(41)] # G. C. Greubel, Mar 17 2025

a(n) = Sum_{k=0..n} binomial(2*k,k)*A158495(n-k).

n*a(n) + 6*(1-n)*a(n-1) + 4*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011

Name edited by Matthew House, Nov 08 2015

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A158496 Expansion of (1-4x+x^2)/(1+x^2)^2.

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A158499 Expansion of (1 + sqrt(1-4x))/(2-4x).

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cp's OEIS Frontend

A158496 Expansion of (1-4x+x^2)/(1+x^2)^2.

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A158499 Expansion of (1 + sqrt(1-4*x))/(2-4*x).

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A158499 Expansion of (1 + sqrt(1-4x))/(2-4x).