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Expansion of ((1-2x)*sqrt(1+2x) + sqrt(1-2x))/(2*(1-2x)^(5/2)).
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%I #18 Nov 30 2022 10:02:12

%S 1,4,11,28,67,156,354,792,1747,3820,8278,17832,38174,81368,172644,

%T 365104,769411,1617228,3389838,7090440,14797546,30828424,64106716,

%U 133113168,275967022,571415416,1181585564,2440680592,5035637212

%N Expansion of ((1-2x)*sqrt(1+2x) + sqrt(1-2x))/(2*(1-2x)^(5/2)).

%C a(n) = Sum_{k=0..n} (k+1)*binomial(n,(n-k)/2)*binomial(k+3,3)*(1+(-1)^(n-k))/(n+k+2). The g.f. is transformed to 1/(1-x)^4 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)^2, where c(x) is the g.f. of the Catalan numbers A000108.

%C 0, 1, 4, 11, 28, ... is the image of the quarter-squares floor((n+1)^2/4) (A002620(n+1)) under the Riordan array ((1+2x)/sqrt(1-4x^2), x*c(x^2)). Hankel transform of A099326 has g.f. (1-x)/(1+x)^4. - _Paul Barry_, Oct 25 2007

%H Vincenzo Librandi, <a href="/A099326/b099326.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = Sum_{k=0..n} (k+1)*binomial(n, (n-k)/2)*binomial(k+3, 3)*(1 + (-1)^(n-k))/(n+k+2).

%F a(n) = Sum_{k=0..n} C(n,k)*(floor((abs(n-2k) + 1)^2/4) + floor((abs(n-2k+1) + 1)^2/4)). - _Paul Barry_, Oct 25 2007

%F D-finite with recurrence: n*(n-2)*a(n) +2*(-n^2+3)*a(n-1) -4*(n-1)*(n-4)*a(n-2) +8*(n-1)*(n-2)*a(n-3)=0. - _R. J. Mathar_, Nov 24 2012

%F a(n) ~ n * 2^(n-1) * (1 + 2*sqrt(2/(Pi*n))). - _Vaclav Kotesovec_, Feb 12 2014

%t CoefficientList[Series[((1-2*x)*Sqrt[1+2*x]+Sqrt[1-2*x])/(2*(1-2*x)^(5/2)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 12 2014 *)

%Y Cf. A099325, A099327.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Oct 12 2004