%I #23 Sep 08 2022 08:46:09
%S 0,1,1,3,9,33,129,531,2265,9921,44361,201651,929073,4328865,20362137,
%T 96562659,461169873,2216134401,10707788721,51988771107,253515373305,
%U 1241069449377,6097106216529,30050252046195,148541591990505,736237012296897
%N Expansion of -(sqrt(x^4-4*x^3-6*x^2-4*x+1) +x^2-2*x-1)/4.
%H Robert Israel, <a href="/A247195/b247195.txt">Table of n, a(n) for n = 0..1391</a>
%F a(n) = Sum_{k=1..n-1} ( Sum_{i=0..n-k-1} 2^i*binomial(k,n-k-i-1)* binomial(k+i-1,k-1) )*binomial(n-k-1,k-1))/k, n>1, a(0)=0, a(1)=1.
%F D-finite with recurrence (n-2)*a(n)+(2-4*n)*a(n+1)+(-6-6*n)*a(n+2)+(-10-4*n)*a(n+3)+(n+4)*a(n+4)=0. - _Robert Israel_, Nov 26 2018
%p f:= gfun:-rectoproc({(n-2)*a(n)+(2-4*n)*a(n+1)+(-6-6*n)*a(n+2)+(-10-4*n)*a(n+3)+(n+4)*a(n+4)=0,a(0) = 0, a(1) = 1, a(2) = 1, a(3) = 3, a(4) = 9},a(n),remember):
%p map(f, [$0..30]); # _Robert Israel_, Nov 26 2018
%t a[n_] := If[n == 1, 1, Sum[Sum[2^i*Binomial[k, n-k-i-1]*Binomial[k+i-1, k-1], {i, 0, n-k-1}]*Binomial[n-k-1, k-1]/k, {k, 1, n-1}]]; Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Nov 25 2014, translated from Maxima *)
%t CoefficientList[Series[-(Sqrt[x^4-4*x^3-6*x^2-4*x+1] + x^2-2*x-1)/4, {x, 0, 30}], x] (* _G. C. Greubel_, Nov 26 2018 *)
%o (Maxima)
%o a(n):=if n=1 then 1 else sum(((sum(2^i*binomial(k,n-k-i-1)*binomial(k+i-1,k-1),i,0,n-k-1))*binomial(n-k-1,k-1))/k,k,1,n-1);
%o (PARI) my(x='x+O('x^30)); concat([0], Vec(-(sqrt(x^4-4*x^3-6*x^2-4*x+1) +x^2-2*x-1)/4)) \\ _G. C. Greubel_, Nov 26 2018
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); [0] cat Coefficients(R!( -(Sqrt(x^4-4*x^3-6*x^2-4*x+1) + x^2-2*x-1)/4 )); // _G. C. Greubel_, Nov 26 2018
%o (Sage) s=(-(sqrt(x^4-4*x^3-6*x^2-4*x+1) +x^2-2*x-1)/4).series(x,30); s.coefficients(x, sparse=False) # _G. C. Greubel_, Nov 26 2018
%K nonn
%O 0,4
%A _Vladimir Kruchinin_, Nov 24 2014
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