%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