A104555 Expansion of x*(1 - x)/(1 - x + x^2)^3.

0, 1, 2, 0, -5, -7, 0, 12, 15, 0, -22, -26, 0, 35, 40, 0, -51, -57, 0, 70, 77, 0, -92, -100, 0, 117, 126, 0, -145, -155, 0, 176, 187, 0, -210, -222, 0, 247, 260, 0, -287, -301, 0, 330, 345, 0, -376, -392, 0, 425, 442, 0, -477, -495

Offset: 0

Author

Paul Barry, Mar 14 2005

Keywords

• easy
• sign

Comments

Image of C(n+1,2) under the Riordan array (1, x*(1-x)).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-6,7,-6,3,-1).

Crossrefs

Cf. A076118, A095130.

Programs

Magma

R:=PowerSeriesRing(Integers(), 60); [0] cat Coefficients(R!( x*(1-x)/(1-x+x^2)^3 )); // G. C. Greubel, Jan 01 2023

Maple

S:=(j,n)->sum(k^j,k=1..n):seq((S(5,n+1)mod S(3,n+1))/((n+1)*(n+2))*(-1)^floor((n mod 6)/4), n=1..40). # Gary Detlefs, Oct 31 2011

Mathematica

CoefficientList[Series[x*(1-x)/(1-x+x^2)^3, {x,0,60}], x]  (* Harvey P. Dale, Apr 13 2011 *)

Sage

def A104555_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1-x)/(1-x+x^2)^3 ).list()
A104555_list(60) # G. C. Greubel, Jan 01 2023

Formula

a(n) = 3*a(n-1) - 6*a(n-2) + 7*a(n-3) - 6*a(n-4) + 3*a(n-5) - a(n-6).

a(n) = Sum_{k=0..n} binomial(k, n-k)(-1)^(n-k)*k(k+1)/2.

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)(-1)^k*(n-k)(n-k+1)/2.

a(3*n) = 0, a(3*n-2) = n*(3*n - 1)/2, a(3*n-1) = n*(3*n + 1)/2. - Ralf Stephan, May 20 2007

a(n) = ((Sum_{k=1..n+1} k^5) mod (Sum_{k=1..n+1} k^3))/((n+1)*(n+2))*(-1)^floor((n mod 6)/4). - Gary Detlefs, Oct 31 2011