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%I #23 Dec 05 2022 04:47:23
%S 1,5,16,45,117,291,700,1646,3799,8647,19448,43330,95738,210094,458216,
%T 994204,2146955,4617439,9893376,21128058,44982486,95510090,202278376,
%U 427425860,901236582,1896594966,3983929680,8354539156,17492095604
%N Expansion of ((1-x)*sqrt(1+2x) + (1+x)*sqrt(1-2x))/(2*(1-2x)^(5/2)).
%C 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.
%H Vincenzo Librandi, <a href="/A099327/b099327.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{k=0..n} (k+1)*binomial(n, (n-k)/2)*binomial(k+4, 4)*(1+(-1)^(n-k))/(n+k+2).
%F D-finite with recurrence: 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
%F a(n) ~ 2^(n+1/2) *n^(3/2) / (3*sqrt(Pi)) * (1 + 9/8*sqrt(2*Pi/n)). - _Vaclav Kotesovec_, Feb 08 2014
%t 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 *)
%Y Cf. A099325, A099326.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Oct 12 2004
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