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A226881
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Number of n-length binary words w with #(w,0) >= #(w,1) >= 1, where #(w,x) gives the number of digits x in w.
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4
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0, 0, 2, 3, 10, 15, 41, 63, 162, 255, 637, 1023, 2509, 4095, 9907, 16383, 39202, 65535, 155381, 262143, 616665, 1048575, 2449867, 4194303, 9740685, 16777215, 38754731, 67108863, 154276027, 268435455, 614429671, 1073741823, 2448023842, 4294967295, 9756737701
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OFFSET
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0,3
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COMMENTS
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a(n) is the number of nonempty subsets of {1,2,...,n} that contain either more even than odd numbers or the same number of even and odd numbers. For example, for n=5, a(5)=15 and the 15 subsets are {2}, {4}, {1,2}, {1,4}, {2,3}, {2,4}, {2,5}, {3,4}, {4,5}, {1,2,4}, {2,3,4}, {2,4,5}, {1,2,3,4}, {1,2,4,5}, {2,3,4,5}. - Enrique Navarrete, Dec 15 2019
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LINKS
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FORMULA
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G.f.: (3*x-1)/(2*(x-1)*(2*x-1)) + 1/(2*sqrt((1+2*x)*(1-2*x))).
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EXAMPLE
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a(4) = 10: 0001, 0010, 0011, 0100, 0101, 0110, 1000, 1001, 1010, 1100.
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MAPLE
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a:= proc(n) option remember;
`if`(n<4, n*(n-1)*(4-n)/2, (9*(n-1)*(n-4) *a(n-1)
+(12-32*n+6*n^2) *a(n-2) -36*(n-2)*(n-4) *a(n-3)
+8*(n-3)*(3*n-10) *a(n-4))/ (n*(3*n-13)))
end:
seq(a(n), n=0..40);
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MATHEMATICA
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Table[Sum[Binomial[n, i], {i, Floor[n/2]}], {n, 0, 30}] (* Wesley Ivan Hurt, Mar 14 2015 *)
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PROG
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(PARI) a(n) = sum(i=1, n\2, binomial(n, i)); \\ Michel Marcus, Jul 15 2022
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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