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

%I #11 Oct 04 2024 05:44:22

%S 1,0,1,3,1,10,6,21,36,43,127,139,340,540,881,1832,2653,5427,8829,

%T 15550,28642,46805,87756,147575,262751,465591,797864,1437816,2471553,

%U 4383696,7689305,13402819,23752217,41305842,72916606,127708213,223809012,394045411

%N Expansion of (1 - x^2 + x^3)/((1 - x^2 + x^3)^2 - 4*x^3).

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,2,-1,2,-1).

%F a(n) = 2*a(n-2) + 2*a(n-3) - a(n-4) + 2*a(n-5) - a(n-6).

%F a(n) = Sum_{k=0..floor(n/2)} binomial(2*k+1,2*n-4*k).

%o (PARI) my(N=40, x='x+O('x^N)); Vec((1-x^2+x^3)/((1-x^2+x^3)^2-4*x^3))

%o (PARI) a(n) = sum(k=0, n\2, binomial(2*k+1, 2*n-4*k));

%Y Cf. A376716, A376788.

%Y Cf. A376723, A376726, A376729.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Oct 04 2024