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

%I A193146 #34 Dec 06 2024 04:04:15
%S A193146 1,1,2,2,4,5,8,10,15,20,29,39,55,75,105,144,200,275,381,525,726,1001,
%T A193146 1383,1908,2635,3636,5020,6928,9564,13200,18221,25149,34714,47914,
%U A193146 66136,91285,126000,173914,240051
%N A193146 Expansion of 1/(1 - x - x^2 + x^3 - x^4 + x^6).
%C A193146 The Gi2 sums, see A180662 for the definition of these sums, of the "Races with Ties" triangle A035317 equal this sequence.
%H A193146 Vincenzo Librandi, <a href="/A193146/b193146.txt">Table of n, a(n) for n = 0..300</a>
%H A193146 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,1,0,-1).
%F A193146 G.f.: 1/(1 - x - x^2 + x^3 - x^4 + x^6).
%F A193146 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.
%F A193146 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.
%F A193146 a(n) = Sum_{k=0..floor(n/2)} binomial(floor(n-3*k/2)+1, n-2*k+1). - _Taras Goy_, Dec 24 2019
%p A193146 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);
%t A193146 CoefficientList[Series[1/(1-x-x^2+x^3-x^4+x^6), {x, 0, 40}], x] (* _Michael De Vlieger_, Dec 24 2019 *)
%t A193146 LinearRecurrence[{1,1,-1,1,0,-1},{1,1,2,2,4,5},50] (* _Harvey P. Dale_, Mar 27 2022 *)
%o A193146 (PARI) Vec(1/(1-x-x^2+x^3-x^4+x^6) +O(x^40)) /* show terms */ \\ _Bruno Berselli_, Jul 22 2011
%o A193146 (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 */
%o A193146 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!(1/(1-x-x^2+x^3-x^4+x^6))); // _Bruno Berselli_, Jul 22 2011
%o A193146 (Sage)
%o A193146 def A193146_list(prec):
%o A193146     P.<x> = PowerSeriesRing(ZZ, prec)
%o A193146     return P( 1/(1-x-x^2+x^3-x^4+x^6) ).list()
%o A193146 A193146_list(40) # _G. C. Greubel_, Jan 01 2020
%o A193146 (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
%Y A193146 Cf. A003269.
%Y A193146 Cf. A035317, A180662.
%K A193146 nonn,easy
%O A193146 0,3
%A A193146 _Johannes W. Meijer_, Jul 20 2011