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A226881 Number of n-length binary words w with #(w,0) >= #(w,1) >= 1, where #(w,x) gives the number of digits x in w. 3

%I

%S 0,0,2,3,10,15,41,63,162,255,637,1023,2509,4095,9907,16383,39202,

%T 65535,155381,262143,616665,1048575,2449867,4194303,9740685,16777215,

%U 38754731,67108863,154276027,268435455,614429671,1073741823,2448023842,4294967295,9756737701

%N Number of n-length binary words w with #(w,0) >= #(w,1) >= 1, where #(w,x) gives the number of digits x in w.

%H Alois P. Heinz, <a href="/A226881/b226881.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: (3*x-1)/(2*(x-1)*(2*x-1)) + 1/(2*sqrt((1+2*x)*(1-2*x))).

%F a(n) = Sum_{i=1..floor(n/2)} C(n,i). - _Wesley Ivan Hurt_, Mar 14 2015

%e a(4) = 10: 0001, 0010, 0011, 0100, 0101, 0110, 1000, 1001, 1010, 1100.

%p a:= proc(n) option remember;

%p `if`(n<4, n*(n-1)*(4-n)/2, (9*(n-1)*(n-4) *a(n-1)

%p +(12-32*n+6*n^2) *a(n-2) -36*(n-2)*(n-4) *a(n-3)

%p +8*(n-3)*(3*n-10) *a(n-4))/ (n*(3*n-13)))

%p end:

%p seq(a(n), n=0..40);

%t Table[Sum[Binomial[n, i], {i, Floor[n/2]}], {n, 0, 30}] (* _Wesley Ivan Hurt_, Mar 14 2015 *)

%Y Column k=2 of A226874.

%K nonn,easy

%O 0,3

%A _Alois P. Heinz_, Jun 21 2013

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Last modified November 13 23:01 EST 2019. Contains 329106 sequences. (Running on oeis4.)