|
%I #15 Aug 14 2024 07:30:49
%S 1,-1,2,0,2,2,5,5,12,16,28,44,73,115,190,304,494,798,1293,2089,3384,
%T 5472,8856,14328,23185,37511,60698,98208,158906,257114,416021,673133,
%U 1089156,1762288,2851444,4613732,7465177,12078907,19544086,31622992,51167078,82790070
%N Expansion of 1/( (1 + x) * (1 - x^2*(1 + x)^2) ).
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (-1,1,3,3,1).
%F a(n) = -a(n-1) + a(n-2) + 3*a(n-3) + 3*a(n-4) + a(n-5).
%F a(n) = Sum_{k=0..floor(n/2)} binomial(2*k-1,n-2*k).
%F a(n) = A375373(n) + A375373(n-1).
%F 2*a(n) = 2*(-1)^n + A000045(n) + A057078(n+1). - _R. J. Mathar_, Aug 14 2024
%o (PARI) my(N=50, x='x+O('x^N)); Vec(1/((1+x)*(1-x^2*(1+x)^2)))
%o (PARI) a(n) = sum(k=0, n\2, binomial(2*k-1, n-2*k));
%Y Cf. A093040, A375373.
%K sign,easy,new
%O 0,3
%A _Seiichi Manyama_, Aug 13 2024
|
