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%I #24 Jan 30 2020 21:29:17
%S 1,1,3,13,59,271,1257,5881,27715,131395,626033,2995147,14380181,
%T 69249337,334345091,1617924973,7844900339,38105139907,185380469961,
%U 903147125143,4405621159969,21515837558557,105188202097091,514747668977263
%N Expansion of (1-2*x)/((2-x)*sqrt(5*x^2-6*x+1))+1/(2-x).
%H G. C. Greubel, <a href="/A262664/b262664.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{k=0..n}(binomial(n,k)*Sum_{i=0..n-k}(2^i*binomial(k,n-k-i)*binomial(k+i-1,i)*(-1)^(n-k-i))).
%F G.f.: A(x) = x*B'(x)/B(x), where B(x)/x is g.f. of A033321.
%F a(n) ~ 5^(n+1/2)/(6*sqrt(Pi*n)). - _Vaclav Kotesovec_, Sep 29 2015
%F D-finite with recurrence: 2*n*(3*n-4)*a(n) = (39*n^2 - 70*n + 28)*a(n-1) - (48*n^2 - 103*n + 34)*a(n-2) + 5*(n-2)*(3*n-1)*a(n-3). - _Vaclav Kotesovec_, Sep 29 2015
%t Table[Sum[Binomial[n, k] Sum[2^i Binomial[k, n - k - i] Binomial[k + i - 1, i] (-1)^(n - k - i), {i, 0, n - k}], {k, 0, n}], {n, 0, 23}] (* _Michael De Vlieger_, Sep 26 2015 *)
%o (Maxima)
%o a(n):=sum(binomial(n,k)*sum(2^i*binomial(k,n-k-i)*binomial(k+i-1,i)*(-1)^(n-k-i),i,0,n-k),k,0,n);
%o (PARI) x='x+O('x^50); Vec((1-2*x)/((2-x)*sqrt(5*x^2-6*x+1))+1/(2-x)) \\ _G. C. Greubel_, Jun 04 2017
%Y Cf. A033321.
%K nonn
%O 0,3
%A _Vladimir Kruchinin_, Sep 26 2015
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