A108474 Expansion of 1/((1-2*x)*(1+4*x^2)).

1, 2, 0, 0, 16, 32, 0, 0, 256, 512, 0, 0, 4096, 8192, 0, 0, 65536, 131072, 0, 0, 1048576, 2097152, 0, 0, 16777216, 33554432, 0, 0, 268435456, 536870912, 0, 0, 4294967296, 8589934592, 0, 0, 68719476736, 137438953472, 0, 0, 1099511627776

Offset: 0

Author

Paul Barry, Jun 04 2005

Keywords

• easy
• nonn

Comments

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

Links

• Index entries for linear recurrences with constant coefficients, signature (2,-4,8).

Formula

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

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

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

a(n) = 2^n*A133872(n). - R. J. Mathar, Mar 08 2021