A097141 Expansion of x*(1+2*x)/(1+x)^2.

0, 1, 0, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 24, -25, 26, -27, 28, -29, 30, -31, 32, -33, 34, -35, 36, -37, 38, -39, 40, -41, 42, -43, 44, -45, 46, -47, 48, -49, 50, -51, 52, -53, 54, -55, 56, -57, 58, -59, 60

Offset: 0

Author

Paul Barry, Jul 29 2004

Keywords

• easy
• sign

Comments

Partial sums of A097140.

Binomial transform is x(1+x)/(1-x), or {0,1,2,2,2,2,....}.

Second binomial transform is x/((1-x)^2(1 - 2x)), or Eulerian numbers A000295(n+1).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-2,-1).

Crossrefs

Cf. A000295, A038608, A040000, A097140, A099570.

Programs

Magma

[0] cat [(n-2)*(-1)^n : n in [1..100]]; // Wesley Ivan Hurt, Dec 11 2016

Maple

A097141:=n->(n-2)*(-1)^n: 0, seq(A097141(n), n=1..100); # Wesley Ivan Hurt, Dec 11 2016

Mathematica

CoefficientList[Series[x (1 + 2 x)/(1 + x)^2, {x, 0, 100}], x] (* Vincenzo Librandi, Mar 11 2014 *)

PARI

a(n)=if(n, (n-2)*(-1)^n, 0) \\ Charles R Greathouse IV, Dec 13 2016

Formula

G.f.: x*(1+2*x)/(1+x)^2.

a(n) = (n-2)*(-1)^n + 2*0^n.

a(n) = -2*a(n-1) - a(n-2) for n > 2.

a(n) = A099570(n) for n > 1. - R. J. Mathar, Dec 15 2008

a(n) = (Sum_{k=1..n} k*(-1)^(n-k)*binomial(n-1,k-1)*binomial(2*n-k-1,n-1))/n, n>0, a(0)=0. - Vladimir Kruchinin, Mar 09 2014

a(n) = A038608(n-2) for n > 2. - Georg Fischer, Oct 06 2018

E.g.f.: 2 - exp(-x)*(2 + x). - Stefano Spezia, Mar 07 2023