A110164 Expansion of (1-x^2)/(1+2*x).

1, -2, 3, -6, 12, -24, 48, -96, 192, -384, 768, -1536, 3072, -6144, 12288, -24576, 49152, -98304, 196608, -393216, 786432, -1572864, 3145728, -6291456, 12582912, -25165824, 50331648, -100663296, 201326592, -402653184, 805306368, -1610612736, 3221225472, -6442450944

Offset

0,2

Comments

Diagonal sums of Riordan array ((1-x)/(1+x),x/(1+x)^2), A110162.

The positive sequence with g.f. (1-x^2)/(1-2*x) gives the row sums of the Riordan array (1+x,x/(1-x)). - Paul Barry, Jul 18 2005

The inverse g.f. is (1 + 2*x + x^2 + 2*x^3 + x^4 + 2*x^5 + x^6 + ...). - Gary W. Adamson, Jan 07 2011

In absolute value, essentially the same as A007283(n) = A003945(n+1) = A042950(n+1) = A082505(n+1) = A087009(n+3) = A091629(n) = A098011(n+4) = A111286(n+2). - M. F. Hasler, Apr 19 2015

Links

Table of n, a(n) for n=0..33.

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

Formula

From M. F. Hasler, Apr 19 2015; (Start)

a(n) = 3*(-2)^(n-2) = 3*A122803(n-2) for n >= 2.

a(n) = -2 a(n-1) for n >= 3. (End)

E.g.f.: (1/4) - (x/2) + (3/4)*exp(-2*x). - Alejandro J. Becerra Jr., Jan 29 2021

From Amiram Eldar, Feb 17 2026: (Start)

Sum_{n>=0} 1/a(n) = 8/9.

Sum_{n>=0} (-1)^n/a(n) = 8/3. (End)

a(n) = Sum_{k=0..n} (-1)^k * A105474(n,k). - Alois P. Heinz, Apr 21 2026

Mathematica

CoefficientList[Series[(1 - x^2)/(1 + 2x), {x, 0, 33}], x] (* Robert G. Wilson v, Jul 08 2006 *)
LinearRecurrence[{-2}, {1, -2, 3}, 40] (* Harvey P. Dale, May 10 2023 *)

Prog

(PARI) A110164(n)=if(n>1, 3*(-2)^(n-2), 1-3*n) \\ M. F. Hasler, Apr 19 2015

Crossrefs

Cf. A003945, A007283, A042950, A082505, A087009, A091629, A098011, A105474, A110162, A111286, A122803.

Sequence in context: A251752 A251766 A042950 * A098011 A367222 A035055

Adjacent sequences: A110161 A110162 A110163 * A110165 A110166 A110167

Keyword

easy,sign

Author

Paul Barry, Jul 14 2005

Status

approved