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A099327 Expansion of ((1-x)sqrt(1+2x)+(1+x)sqrt(1-2x))/(2(1-2x)^(5/2)). 3
1, 5, 16, 45, 117, 291, 700, 1646, 3799, 8647, 19448, 43330, 95738, 210094, 458216, 994204, 2146955, 4617439, 9893376, 21128058, 44982486, 95510090, 202278376, 427425860, 901236582, 1896594966, 3983929680, 8354539156, 17492095604 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The g.f. is transformed to 1/(1-x)^5 under the Chebyshev transformation A(x)->1/(1+x^2)A(x/(1+x^2)). Second binomial transform of the sequence with g.f. 1/c(-x)^3, where c(x) is the g.f. of the Catalan numbers A000108.

LINKS

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

FORMULA

a(n)=sum{k=0..n, (k+1)binomial(n, (n-k)/2)binomial(k+4, 4)(1+(-1)^(n-k))/(n+k+2)}.

Conjecture: +n*(n-3)*a(n) +2*(-n^2+6)*a(n-1) +4*-(n-1)*(n-5)*a(n-2) +8*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Nov 24 2012

a(n) ~ 2^(n+1/2) *n^(3/2) / (3*sqrt(Pi)) * (1 + 9/8*sqrt(2*Pi/n)). - Vaclav Kotesovec, Feb 08 2014

MATHEMATICA

CoefficientList[Series[((1-x)*Sqrt[1+2*x]+(1+x)*Sqrt[1-2*x])/(2*(1-2*x)^(5/2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 08 2014 *)

CROSSREFS

Cf. A099325, A099326.

Sequence in context: A282425 A185003 A189390 * A004146 A275126 A071101

Adjacent sequences:  A099324 A099325 A099326 * A099328 A099329 A099330

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Oct 12 2004

STATUS

approved

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Last modified April 1 05:04 EDT 2020. Contains 333155 sequences. (Running on oeis4.)