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A360705 Expansion of Sum_{k>=0} (x * (1 + (-1)^k * x))^k.

%I A360705 #25 Mar 12 2023 10:34:03
%S A360705 1,1,0,3,-1,8,1,21,0,55,-1,144,1,377,0,987,-1,2584,1,6765,0,17711,-1,
%T A360705 46368,1,121393,0,317811,-1,832040,1,2178309,0,5702887,-1,14930352,1,
%U A360705 39088169,0,102334155,-1,267914296,1,701408733,0,1836311903,-1,4807526976,1
%N A360705 Expansion of Sum_{k>=0} (x * (1 + (-1)^k * x))^k.
%H A360705 Winston de Greef, <a href="/A360705/b360705.txt">Table of n, a(n) for n = 0..4762</a>
%H A360705 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,1,0,2,0,-1)
%F A360705 a(n) = Sum_{k=0..floor(n/2)} (-1)^(k*(n-k)) * binomial(n-k,k).
%F A360705 a(2*n) = A057078(n), a(2*n+1) = A000045(2*n+2).
%F A360705 G.f.: ( 1+x+x^3-2*x^4+x^5+x^6-2*x^2 ) / ( (x^2-x-1)*(x^2+x-1)*(1+x+x^2)*(x^2-x+1) ). - _R. J. Mathar_, Mar 12 2023
%o A360705 (PARI) my(N=50, x='x+O('x^N)); Vec(sum(k=0, N, (x*(1+(-1)^k*x))^k))
%o A360705 (PARI) a(n) = sum(k=0, n\2, (-1)^(k*(n-k))*binomial(n-k, k));
%o A360705 (PARI) a(n) = if(n%2, fibonacci(n+1), [1, 0, -1][n/2%3+1]);
%Y A360705 Cf. A360592, A360696, A360704.
%Y A360705 Cf. A000045, A001906, A057078.
%K A360705 sign,easy
%O A360705 0,4
%A A360705 _Seiichi Manyama_, Feb 17 2023