A178945 Expansion of x*(1-x)^2/( (1-2*x^2)*(1-2*x)^2).

1, 2, 7, 16, 42, 96, 228, 512, 1160, 2560, 5648, 12288, 26656, 57344, 122944, 262144, 557184, 1179648, 2490624, 5242880, 11010560, 23068672, 48235520, 100663296, 209717248, 436207616, 905973760, 1879048192, 3892322304, 8053063680, 16643014656

Offset

1,2

Comments

Let S(x) be the generating function for A000079. Then the generating function for this sequence is x(S(x)^2+S(x^2))/2.

Links

Table of n, a(n) for n=1..31.

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

Formula

a(2n+1) = ( A001787(2n+1)+A077957(2n))/2.

a(2n) = A001787(2n)/2.

a(n) = 2^(n-2)*n + 2^(n/2-5/2)*(1-(-1)^n).

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

Example

(1, 4, 12, 32, 80, 192, 448, 1024,...) +
..(1, 0,..2,..0,..4,...0,...8,....0...) =
..(2, 4, 14, 32, 84, 192, 456, 1024,...). Then dividing the sum by 2 we obtain:
..(1, 2, 7, 16, 42, 96, 228,...).

Mathematica

CoefficientList[Series[x (1-x)^2/((1-2x^2)(1-2x)^2), {x, 0, 50}], x] (* or *) LinearRecurrence[{4, -2, -8, 8}, {0, 1, 2, 7}, 50] (* Harvey P. Dale, Dec 29 2023 *)

Crossrefs

Cf. A001787, A077957, column k=2 of A290222.

Sequence in context: A320236 A073371 A113224 * A309561 A026571 A100099

Adjacent sequences: A178942 A178943 A178944 * A178946 A178947 A178948

Keyword

nonn,easy

Author

Gary W. Adamson, Dec 30 2010

Status

approved