A128058 -id:A128058 - cp's OEIS frontend

A128056 Hankel transform of A128057.

1, -3, -6, 28, 56, -288, -576, 3008, 6016, -31488, -62976, 329728, 659456, -3452928, -6905856, 36159488, 72318976, -378667008, -757334016, 3965452288, 7930904576, -41526755328, -83053510656, 434873827328

Offset: 0

Paul Barry, Feb 13 2007

a(n)=2^A128054(n)*A128053(n). Hankel transform of A128058.

a(n)=A128055(n)*((cos(pi*n/2)-sin(pi*n/2))((F(n-1)+F(n+1))(5/6-cos(pi*n/3)/3)(1+(-1)^n)/2 +(F(n)+F(n+2))(5/6-cos(pi*(n+1)/3)/3)(1-(-1)^n)/2)).

Empirical g.f.: -(2*x-1)*(4*x^2-x+1) / (16*x^4+12*x^2+1). - Colin Barker, Jun 27 2013

A128057 Expansion of (1+x)/sqrt(1+4x^2).

Original entry on oeis.org

1, 1, -2, -2, 6, 6, -20, -20, 70, 70, -252, -252, 924, 924, -3432, -3432, 12870, 12870, -48620, -48620, 184756, 184756, -705432, -705432, 2704156, 2704156, -10400600, -10400600, 40116600, 40116600, -155117520, -155117520

Offset: 0

Paul Barry, Feb 13 2007

Hankel transform is A128056. Binomial transform is A128058. Unsigned version is A128014.

Cf. A000984.

CoefficientList[Series[(1+x)/Sqrt[1+4x^2],{x,0,40}],x] (* Harvey P. Dale, May 12 2015 *)

a(n)=(-1)^C(n,2)*(C(n,n/2)*(1+(-1)^n)/2+C(n-1,(n-1)/2)*(1-(-1)^n)/2);
Conjecture: n*(5*n-9)*a(n) +4*a(n-1) +4*(5*n-4)*(n-2)*a(n-2)=0. - R. J. Mathar, Dec 02 2014
D-finite with recurrence: n*a(n) +(n-2)*a(n-1) +4*(n-1)*a(n-2) +4*(n-3)*a(n-3)=0. - R. J. Mathar, Dec 02 2014

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A128056 Hankel transform of A128057.

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A128057 Expansion of (1+x)/sqrt(1+4x^2).

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