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A100097 An inverse Chebyshev transform of the Pell numbers. 4
0, 1, 2, 8, 20, 64, 172, 512, 1416, 4096, 11468, 32768, 92248, 262144, 739832, 2097152, 5925520, 16777216, 47429900, 134217728, 379536440, 1073741824, 3036661032, 8589934592, 24294699120, 68719476736, 194363001272 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Image of x/(1-2x-x^2) under the transform g(x)->(1/sqrt(1-4x^2)g(xc(x^2)), where c(x) is the g.f. of the Catalan numbers A000108. This is the inverse of the Chebyshev transform which takes A(x) to ((1-x^2)/(1+x^2))A(x/(1+x^2)).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

FORMULA

G.f.: x*sqrt(1-4*x^2)*(sqrt(1-4*x^2)+2*x)/((1-4*x^2)*(1-8*x^2)).

a(n) = sum{k=0..floor(n/2), binomial(n, k)*A000129(n-2k)}.

Conjecture: (-n+2)*a(n) +(-n+3)*a(n-1) +4*(3*n-7)*a(n-2) +4*(3*n-10)*a(n-3) +32*(-n+3)*a(n-4) +32*(-n+4)*a(n-5)=0. - R. J. Mathar, Nov 24 2012

Recurrence: (n-2)*a(n) = 4*(3*n-7)*a(n-2) - 32*(n-3)*a(n-4). - Vaclav Kotesovec, Feb 12 2014

a(n) ~ 2^((3*n-3)/2). - Vaclav Kotesovec, Feb 12 2014

a(2*n) = 8^n/(2*sqrt(2)) - 2^n * (2*n-1)!! * hypergeom([1, n+1/2], [n+1], 1/2)/(4*n!), a(2*n+1) = 8^n. - Vladimir Reshetnikov, Oct 13 2016

MATHEMATICA

CoefficientList[Series[x*Sqrt[1-4*x^2]*(Sqrt[1-4*x^2]+2*x)/((1-4*x^2)*(1-8*x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)

CROSSREFS

Cf. A100095, A100096.

Sequence in context: A024997 A081157 A099177 * A133467 A091004 A005559

Adjacent sequences:  A100094 A100095 A100096 * A100098 A100099 A100100

KEYWORD

nonn,easy

AUTHOR

Paul Barry, Nov 03 2004

STATUS

approved

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Last modified January 22 16:37 EST 2020. Contains 331152 sequences. (Running on oeis4.)