%I #27 May 07 2021 16:54:47
%S 0,1,6,6,-100,-570,-588,8092,45432,47430,-607420,-3385932,-3557112,
%T 43868188,243513480,256815480,-3094459408,-17130508218,-18113603868,
%U 214848211780,1187079671400,1257576694836,-14747640408424,-81367084566264,-86322262278000,1003635505135900
%N Expansion of x/ (1-4*x+16*x^2)^(3/2).
%F a(n)= (2(2n-1)/(n-1))*a(n-1) - (16n/(n-1))*a(n-2), starting with a(0) = 0 and a(1) = 1. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 15 2004
%F For n > 0, a(n) = (n+1)*A025175(n-1)/2. - _Vladimir Reshetnikov_, Nov 01 2015
%p A012125:=proc(n) options remember: if n<2 then RETURN([0,1][n+1]) else RETURN((2*(2*n-1)/(n-1))*procname(n-1)-(16*n/(n-1))*procname(n-2)) fi: end; seq(A012125(n), n=0..25);
%p seq(coeff(convert(series(x/((1-4*x+16*x^2)^(3/2)),x,40),polynom),x,i), i=0..25); # C. Ronaldo
%t Table[ -((2^(-1 + 2*n)*LegendreP[ n, 1, 1/2 ])/Sqrt[ 3 ]), {n, 0, 12} ]
%t Table[4^n (n+1) (LegendreP[n, 1/2] - 2 LegendreP[n+1, 1/2])/6, {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 01 2015 *)
%t CoefficientList[Series[x/(1-4x+16x^2)^(3/2),{x,0,30}],x] (* _Harvey P. Dale_, May 07 2021 *)
%o (PARI) x='x+O('x^50); concat(0, Vec(x/(1-4*x+16*x^2)^(3/2))) \\ _Altug Alkan_, Nov 02 2015
%o (PARI) a(n) = 4^n*(n+1)*(pollegendre(n, 1/2) -2*pollegendre(n+1, 1/2))/6; \\ _Michel Marcus_, Nov 03 2015
%Y Cf. A025175.
%K sign
%O 0,3
%A _Wouter Meeussen_
%E Simpler definition and more terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 15 2004