A120617 Hankel transform of g.f. 1/sqrt(1+4x^2).

1, -2, -4, 8, 16, -32, -64, 128, 256, -512, -1024, 2048, 4096, -8192, -16384, 32768, 65536, -131072, -262144, 524288, 1048576, -2097152, -4194304, 8388608, 16777216, -33554432, -67108864, 134217728, 268435456, -536870912, -1073741824, 2147483648, 4294967296, -8589934592

Offset: 0

Author

Paul Barry, Jun 17 2006

Keywords

• sign
• easy

Comments

Hankel transform of e.g.f. Bessel_I(0,2*sqrt(-1)*x) or (1,0,-2,0,6,0,-20,...).

Hankel transform of Sum{k=0..n} (-1)^(n-k)*C(n,k)^2.

Hankel transform of A098331.

Hankel transform of A082590. - Paul Barry, Apr 26 2009

Links

• Index entries for linear recurrences with constant coefficients, signature (0,-4).

Crossrefs

Cf. A082590, A098331.

Programs

Mathematica

LinearRecurrence[{0,-4},{1,-2},40] (* or *) CoefficientList[ Series[ (1-2x)/(1+4x^2),{x,0,40}],x] (* Harvey P. Dale, Oct 12 2011 *)

Formula

G.f.: (1-2*x)/(1+4*x^2); a(n) = 2^n*(cos(Pi*(n+1)/2)+sin(Pi*(n+1)/2)).

a(0)=1, a(1)=-2, a(n)=-4*a(n-2). - Harvey P. Dale, Oct 12 2011

a(n) = ( 2*i^(n+1) )^n, where i=sqrt(-1). - Bruno Berselli, Oct 12 2011

E.g.f.: cos(2*x) - sin(2*x). - Arkadiusz Wesolowski, Aug 31 2012