A395396 - OEIS

%I #13 Apr 27 2026 17:32:49

%S -1,0,0,2,4,9,18,34,68,125,250,461,922,1715,3430,6434,12868,24309,

%T 48618,92377,184754,352715,705430,1352077,2704154,5200299,10400598,

%U 20058299,40116598,77558759,155117518,300540194,601080388,1166803109,2333606218,4537567649,9075135298,17672631899

%N a(n) = binomial(n,floor(n/2)) + (n mod 2) - 2.

%C For n > 0, number of proper, nonempty subsets of {1,2,...,n} containing as many even numbers as odd numbers.

%F G.f.: (2*x+1-sqrt(1-4*x^2))/(2*x*sqrt(1-4*x^2)) + x/(1-x^2) - 2/(1-x).

%F a(n) ~ 2^(n+1/2)/sqrt(n*Pi). - _Stefano Spezia_, Apr 21 2026

%F D-finite with recurrence: (4 + 4*n)*a(n) + (18 + 8*n)*a(n + 1) + (13 + 3*n)*a(n + 2) + (-2*n - 6)*a(n + 3) + (-5 - n)*a(n + 4) + 36 + 18*n = 0. - _Robert Israel_, Apr 22 2026

%p f:= gfun:-rectoproc({(4 + 4*n)*a(n) + (18 + 8*n)*a(n + 1) + (13 + 3*n)*a(n + 2) + (-2*n - 6)*a(n + 3) + (-5 - n)*a(n + 4) + 36 + 18*n, a(0) = -1, a(1) = 0, a(2) = 0, a(3) = 2, a(4) = 4}, a(n), remember):

%p map(f, [$0..50]); # _Robert Israel_, Apr 22 2026

%t a[n_]:=Binomial[n,Floor[n/2]] + Mod[n,2] - 2; Array[a,38,0] (* _Stefano Spezia_, Apr 21 2026 *)

%Y Cf. A001405, A014495.

%K sign,easy

%O 0,4

%A _Enrique Navarrete_, Apr 21 2026