|
%I #20 Feb 06 2024 08:13:53
%S 1,2,7,24,87,322,1211,4604,17645,68042,263655,1025632,4002601,
%T 15662422,61427543,241386924,950160607,3745589510,14784496003,
%U 58424093536,231112008371,915065382154,3626113490579,14379912928572,57064644495359
%N Expansion of 1/(sqrt(1-4*x) - x^2).
%C Diagonal sums of convolution triangle of central binomial coefficients A054335.
%C Number of lattice paths from (0,0) to (n,n) with steps (0,1), (1,0) and, when on the diagonal, (2,2). - _Alois P. Heinz_, Sep 14 2016
%H Alois P. Heinz, <a href="/A104625/b104625.txt">Table of n, a(n) for n = 0..1000</a>
%F Conjecture: n*a(n) + (n-3)*a(n-1) + 2*(-28*n+51)*a(n-2) + 72*(2*n-5)*a(n-3) - n*a(n-4) + (-5*n+3)*a(n-5) + 18*(2*n-5)*a(n-6) = 0. - _R. J. Mathar_, Feb 20 2015
%F a(n) = Sum_{k=0..floor(n+2)/2)} 4^(n+2-2*k) * binomial(n+1-3*k/2,n+2-2*k). - _Seiichi Manyama_, Feb 06 2024
%t CoefficientList[Series[1/(Sqrt[1-4*x] -x^2), {x, 0, 50}], x] (* _G. C. Greubel_, Aug 12 2018 *)
%o (PARI) x='x+O('x^50); Vec(1/(sqrt(1-4*x) - x^2)) \\ _G. C. Greubel_, Aug 12 2018
%o (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1/(sqrt(1-4*x) - x^2))); // _G. C. Greubel_, Aug 12 2018
%Y Cf. A000984, A026671, A054335.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Mar 17 2005
|
