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

0,3

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).

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

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 *)

Prog

(PARI) Vec(1/(1-x-x^2+x^3-x^4+x^6) +O(x^40)) /* show terms */ \\ Bruno Berselli, Jul 22 2011
(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 */
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!(1/(1-x-x^2+x^3-x^4+x^6))); // Bruno Berselli, Jul 22 2011
(Sage)
def A193146_list(prec):
    P.<x> = 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
(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

Crossrefs

Cf. A003269.

Cf. A035317, A180662.

Sequence in context: A325434 A326631 A260894 * A240486 A237871 A027193

Adjacent sequences: A193143 A193144 A193145 * A193147 A193148 A193149

Keyword

nonn,easy

Author

Johannes W. Meijer, Jul 20 2011

Status

approved