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%I #24 Sep 08 2022 08:46:09
%S 0,2,3,2,10,11,21,50,66,152,275,467,988,1717,3283,6386,11560,22556,
%T 42465,79832,154122,290039,554323,1060259,2012310,3859286,7365423,
%U 14072333,26980788,51580271,98873291,189567090,363277676,697348910
%N Expansion of (-3/2+(x^3+3*x)/(sqrt(x^4-4*x^3-2*x^2+1)*2*x)).
%H Harvey P. Dale, <a href="/A247170/b247170.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = n*Sum_{k=1..n} binomial(k,n-2*k)*binomial(n-k-1,k-1)/k.
%F From _R. J. Mathar_, Jan 25 2020: (Start)
%F D-finite with recurrence: +3*n*a(n) +3*(n-1)*a(n-1) +(-5*n+2)*a(n-2) +(-17*n+25)*a(n-3) +(-11*n+34)*a(n-4) +(-3*n+25)*a(n-5) +(-3*n+20)*a(n-6) +(n-7)*a(n-7) = 0.
%F Conjectured: +n*(2*n-7)*a(n) +(n-1)*(2*n-9)*a(n-1) +2*(-2*n^2+9*n-6)*a(n-2) +2*(-6*n^2+33*n-38)*a(n-3) +3*(-2*n^2+15*n-26)*a(n-4) +(2*n-5)*(n-5)*a(n-5) = 0.
%F (End)
%t Table[n*Sum[(Binomial[k,n-2k]Binomial[n-k-1,k-1])/k,{k,n}],{n,40}] (* _Harvey P. Dale_, Oct 04 2017 *)
%o (Maxima)
%o a(n):=n*sum((binomial(k,n-2*k)*binomial(n-k-1,k-1))/k,k,1,n);
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 36); [0] cat Coefficients(R!( (-3/2+(x^3+3*x)/(Sqrt(x^4-4*x^3-2*x^2+1)*2*x)))); // _Marius A. Burtea_, Feb 11 2020
%o (Magma) [n*&+[Binomial(k,n-2*k)*Binomial(n-k-1,k-1)/k:k in [1..n]]:n in [1..35]]; // _Marius A. Burtea_, Feb 11 2020
%Y Cf. A025250.
%K nonn
%O 1,2
%A _Vladimir Kruchinin_, Nov 21 2014
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