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A122877 Expansion of (1-2*x-3*x^2-(1-x)*sqrt(1-2*x-7*x^2))/(8*x^3). 1

%I

%S 0,1,2,7,20,65,206,679,2248,7569,25690,88055,303964,1056497,3693158,

%T 12977655,45813008,162400609,577843890,2063053991,7388487460,

%U 26535797729,95552015614,344897769991,1247685613272

%N Expansion of (1-2*x-3*x^2-(1-x)*sqrt(1-2*x-7*x^2))/(8*x^3).

%C Binomial transform is A071357.

%H Michael De Vlieger, <a href="/A122877/b122877.txt">Table of n, a(n) for n = 0..1723</a>

%H Georg Fischer, Richard J. Mathar, <a href="http://www.mpia.de/~mathar/public/fischer20010119.pdf">p-finite recurrences from sqrt generating functions</a>, (2020).

%F a(n) = Sum_{k=0..n} C(n,k)*2^((k-1)/2)*C((k-1)/2+1)*(1-(-1)^k)/2, where C(n)=A000108(n).

%F a(n) = (1/Pi)*Integral_{x=1-2*sqrt(2)..1+2*sqrt(2)} x^n*sqrt(-x^2+2x+7)*(x-1)/8.

%F a(n) = (Sum_{j=0..n+1} binomial(j,n-j+3)*2^(n-j+2)*binomial(n+1,j))/(n+1). - _Vladimir Kruchinin_, May 19 2014

%F D-finite with recurrence: (n+3)*a(n) + (-3*n-4)*a(n-1) + (-5*n-1)*a(n-2) + 7*(n-2)*a(n-3) = 0. - _R. J. Mathar_, Feb 23 2015

%F Conjecture: -(n+3)*(n-1)*a(n) + n*(2*n+1)*a(n-1) + 7*n*(n-1)*a(n-2) = 0. - _R. J. Mathar_, Feb 23 2015

%F a(n) ~ (1 + 2*sqrt(2))^(n + 3/2) / (sqrt(Pi) * 2^(5/4) * n^(3/2)). - _Vaclav Kotesovec_, Sep 03 2019

%t CoefficientList[Series[(1 - 2 x - 3 x^2 - (1 - x) Sqrt[1 - 2 x - 7 x^2])/(8 x^3), {x, 0, 24}], x] (* _Michael De Vlieger_, Apr 17 2020 *)

%o (Maxima)

%o a(n):=sum(binomial(j,n-j+3)*2^(n-j+2)*binomial(n+1,j),j,0,n+1)/(n+1); /* _Vladimir Kruchinin_, May 19 2014 */

%K easy,nonn

%O 0,3

%A _Paul Barry_, Sep 16 2006

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Last modified October 27 01:02 EDT 2020. Contains 338035 sequences. (Running on oeis4.)