A012125 Expansion of x/ (1-4*x+16*x^2)^(3/2).

0, 1, 6, 6, -100, -570, -588, 8092, 45432, 47430, -607420, -3385932, -3557112, 43868188, 243513480, 256815480, -3094459408, -17130508218, -18113603868, 214848211780, 1187079671400, 1257576694836, -14747640408424, -81367084566264, -86322262278000, 1003635505135900

Offset

0,3

Links

Table of n, a(n) for n=0..25.

Formula

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

For n > 0, a(n) = (n+1)*A025175(n-1)/2. - Vladimir Reshetnikov, Nov 01 2015

Maple

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);
seq(coeff(convert(series(x/((1-4*x+16*x^2)^(3/2)), x, 40), polynom), x, i), i=0..25); # C. Ronaldo

Mathematica

Table[ -((2^(-1 + 2*n)*LegendreP[ n, 1, 1/2 ])/Sqrt[ 3 ]), {n, 0, 12} ]
Table[4^n (n+1) (LegendreP[n, 1/2] - 2 LegendreP[n+1, 1/2])/6, {n, 0, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)
CoefficientList[Series[x/(1-4x+16x^2)^(3/2), {x, 0, 30}], x] (* Harvey P. Dale, May 07 2021 *)

Prog

(PARI) x='x+O('x^50); concat(0, Vec(x/(1-4*x+16*x^2)^(3/2))) \\ Altug Alkan, Nov 02 2015
(PARI) a(n) = 4^n*(n+1)*(pollegendre(n, 1/2) -2*pollegendre(n+1, 1/2))/6; \\ Michel Marcus, Nov 03 2015

Crossrefs

Cf. A025175.

Sequence in context: A361738 A320824 A085804 * A267139 A170915 A328186

Adjacent sequences: A012122 A012123 A012124 * A012126 A012127 A012128

Keyword

sign

Author

Wouter Meeussen

Extensions

Simpler definition and more terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 15 2004

Status

approved