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%I #7 Feb 03 2024 16:18:31
%S 1,0,2,2,6,8,20,32,70,120,252,452,924,1696,3432,6400,12870,24240,
%T 48620,92252,184756,352464,705432,1351616,2704156,5199376,10400600,
%U 20056584,40116600,77555328,155117520,300533760,601080390,1166790240
%N Expansion of g(x) - x*g(x^2), where g(x) is the g.f. of A001405.
%C Partial sums are A129384.
%H G. C. Greubel, <a href="/A129383/b129383.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: 2/(1-2*x+sqrt(1-4*x^2)) - 2*x/(1-2*x^2+sqrt(1-4*x^4)).
%F a(n) = binomial(n,floor(n/2)) - (1/2)*(1-(-1)^n)*binomial((n-1)/2, floor((n-1)/4)).
%t A129383[n_]:= With[{B=Binomial,F=Floor}, B[n,F[n/2]] - Mod[n,2]*B[(n- 1)/2, F[(n-1)/4]]];
%t Table[A129383[n], {n,0,40}] (* _G. C. Greubel_, Feb 03 2024 *)
%o (Magma)
%o A129383:= func< n | Binomial(n,Floor(n/2)) - (n mod 2)*Binomial(Floor((n-1)/2),Floor((n-1)/4)) >;
%o [A129383(n): n in [0..40]]; // _G. C. Greubel_, Feb 03 2024
%o (SageMath)
%o def A129383(n): return binomial(n,n//2) - (n%2)*binomial((n-1)/2,(n-1)//4)
%o [A129383(n) for n in range(41)] # _G. C. Greubel_, Feb 03 2024
%Y Cf. A001405, A129384.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Apr 12 2007
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