A193146 Expansion of 1/(1 - x - x^2 + x^3 - x^4 + x^6).

1, 1, 2, 2, 4, 5, 8, 10, 15, 20, 29, 39, 55, 75, 105, 144, 200, 275, 381, 525, 726, 1001, 1383, 1908, 2635, 3636, 5020, 6928, 9564, 13200, 18221, 25149, 34714, 47914, 66136, 91285, 126000, 173914, 240051

Offset: 0

Author

Johannes W. Meijer, Jul 20 2011

Keywords

• nonn
• easy

Comments

The Gi2 sums, see A180662 for the definition of these sums, of the "Races with Ties" triangle A035317 equal this sequence.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,0,-1).

Crossrefs

Cf. A003269.

Cf. A035317, A180662.

Programs

GAP

a:=[1,1,2,2,4,5];; for n in [7..40] do a[n]:=a[n-1]+a[n-2]-a[n-3]+a[n-4]-a[n-6]; od; a; # G. C. Greubel, Jan 01 2020

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!(1/(1-x-x^2+x^3-x^4+x^6))); // Bruno Berselli, Jul 22 2011

Maple

A193146 := proc(n) option remember: if n>=-5 and n<=-1 then 0 elif n=0 then 1 else procname(n-1) + procname(n-2) - procname(n-3) + procname(n-4) - procname(n-6) fi: end: seq(A193146(n), n=0..40);

Mathematica

CoefficientList[Series[1/(1-x-x^2+x^3-x^4+x^6), {x, 0, 40}], x] (* Michael De Vlieger, Dec 24 2019 *)
LinearRecurrence[{1,1,-1,1,0,-1},{1,1,2,2,4,5},50] (* Harvey P. Dale, Mar 27 2022 *)

Maxima

makelist(coeff(taylor(1/(1-x-x^2+x^3-x^4+x^6), x, 0, n), x, n), n, 0, 40); /* Bruno Berselli, Jul 22 2011 */

PARI

Vec(1/(1-x-x^2+x^3-x^4+x^6) +O(x^40)) /* show terms */ \\ Bruno Berselli, Jul 22 2011

Sage

def A193146_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-x-x^2+x^3-x^4+x^6) ).list()
A193146_list(40) # G. C. Greubel, Jan 01 2020

Formula

G.f.: 1/(1 - x - x^2 + x^3 - x^4 + x^6).

a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-6) with a(n) = 0 for n = -5, -4, -3, -2, -1 and a(0) = 1.

a(n) = b(n) + b(n-1) + b(n-3) - (1-(-1)^n)/2 with b(n) = A003269(n) and b(-3) = b(-2) = b(-1) = 0.

a(n) = Sum_{k=0..floor(n/2)} binomial(floor(n-3*k/2)+1, n-2*k+1). - Taras Goy, Dec 24 2019