A395396 a(n) = binomial(n,floor(n/2)) + (n mod 2) - 2.

-1, 0, 0, 2, 4, 9, 18, 34, 68, 125, 250, 461, 922, 1715, 3430, 6434, 12868, 24309, 48618, 92377, 184754, 352715, 705430, 1352077, 2704154, 5200299, 10400598, 20058299, 40116598, 77558759, 155117518, 300540194, 601080388, 1166803109, 2333606218, 4537567649, 9075135298, 17672631899

Offset

0,4

Comments

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

Links

Table of n, a(n) for n=0..37.

Formula

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

a(n) ~ 2^(n+1/2)/sqrt(n*Pi). - Stefano Spezia, Apr 21 2026

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

Maple

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):
map(f, [$0..50]); # Robert Israel, Apr 22 2026

Mathematica

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

Crossrefs

Cf. A001405, A014495.

Sequence in context: A046683 A065055 A065030 * A103321 A138196 A298404

Adjacent sequences: A395393 A395394 A395395 * A395397 A395398 A395399

Keyword

sign,easy

Author

Enrique Navarrete, Apr 21 2026

Status

approved