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

%I #18 Jan 01 2024 08:33:06

%S 1,2,0,0,16,32,0,0,256,512,0,0,4096,8192,0,0,65536,131072,0,0,1048576,

%T 2097152,0,0,16777216,33554432,0,0,268435456,536870912,0,0,4294967296,

%U 8589934592,0,0,68719476736,137438953472,0,0,1099511627776

%N Expansion of 1/((1-2x)*(1+4x^2)).

%C 2^n with gaps. In general, Sum_{k=0..n} Sum_{j=0..n} C(2(n-k),j)*C(2k,j)*r^j has expansion (1 - (r+1)*x)/((1 + (r+3)x + (r-1)(r+3)x^2 + (r-1)^3*x^3).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,-4,8).

%F G.f.: 1/(1-2x+4x^2-8x^3);

%F a(n) = 2*a(n-1) - 4*a(n-2) + 8*a(n-3);

%F a(n) = Sum_{k=0..n} Sum_{j=0..n} C(2(n-k), j)*C(2k, j)*(-1)^j.

%F a(n) = 2^n*A133872(n). - _R. J. Mathar_, Mar 08 2021

%K easy,nonn

%O 0,2

%A _Paul Barry_, Jun 04 2005