A298072 Number of binary strings of length n that are "prefix heavy", meaning that the fraction of "1" bits in any nonempty prefix is at least as great as the fraction of "1" bits in the entire string.

1, 2, 3, 4, 6, 8, 15, 20, 40, 64, 126, 188, 434, 632, 1391, 2428, 4826, 7712, 17744, 27596, 62468, 106934, 220288, 364724, 834200, 1384470, 2954760, 5187588, 11085712, 18512792, 42379925, 69273668, 152600856, 268881898, 570336966, 1023023000, 2205306276

Offset: 0

Author

Lee A. Newberg, Jan 11 2018

Keywords

• nonn

Comments

If each "1" is interpreted as a unit step to the north and each "0" is interpreted as a unit step to the east, then a prefix heavy string is any string for which the path taken as the bits of the string are examined from left to right has the property that the path does not go southeast of the line segment connecting the start of the path to the end of the path.

Every Dyck word, that is, every balanced string of parentheses, with "(" identified as "1" and ")" identified as "0", is prefix heavy because the fraction of "1" bits in the entire string is necessarily 0.5 and each prefix of nonzero length has at least half of its positions with a "1" bit. That is, each prefix has at least as many "(" characters as it has ")" characters.

Provably, a(n) is even if n is odd because there is a one-to-one relationship between a prefix heavy string with fewer than n/2 "1" bits and its reverse complement with more than n/2 "1" bits.

Provably, a(n) is odd if and only if n+2 is an integer power of 2. That is, a(n) is odd if and only if there are strings with exactly n/2 "1" bits (n is even) and there are an odd number of them that are prefix heavy (A000108(n/2) is odd).

Examples

			For n=2, the a(2)=3 prefix heavy strings of length 2 are 00, 10, and 11, because each prefix of nonzero length is constituted of at least 0%, 50%, or 100% "1" bits, respectively.
For n=6, the a(6)=15 prefix heavy strings of length 6 are 000000, 100000, 100100, 101000, 101010, 101100, 110000, 110010, 110100, 110110, 111000, 111010, 111100, 111110, and 111111.
		

Links

• Lee A. Newberg, R code and Table of n, a(n) for n = 0..1024

Crossrefs

Cf. A000108, A071201.

Formula

a(n) = 2 + Sum_{k=1..n-1} A071201(n-k,k) for n>0.

a(2n) >= A000108(n) because Dyck words are prefix heavy.

Extensions

a(28)-a(36) from Alois P. Heinz, Jan 11 2018