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

0,3

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).

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

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 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Integers(), 60); [0] cat Coefficients(R!( x*(1-x)/(1-x+x^2)^3 )); // G. C. Greubel, Jan 01 2023
(Sage)
def A104555_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( x*(1-x)/(1-x+x^2)^3 ).list()
A104555_list(60) # G. C. Greubel, Jan 01 2023

Crossrefs

Cf. A076118, A095130.

Sequence in context: A320372 A097709 A197877 * A140571 A078049 A021490

Adjacent sequences: A104552 A104553 A104554 * A104556 A104557 A104558

Keyword

easy,sign

Author

Paul Barry, Mar 14 2005

Status

approved