

A164953


Square array read by antidiagonals: a(m,n) = the number of different multisets (in respect to the value of the digits) of lengths of runs in the binary representations of positive integers that contain exactly m 0's and n 1's in binary. (The leftmost digit must be 1 in each binary number.)


0



1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 3, 4, 3, 1, 1, 4, 5, 6, 3, 1, 1, 4, 7, 8, 7, 4, 1, 1, 5, 8, 11, 10, 10, 4, 1, 1, 5, 11, 14, 15, 15, 11, 5, 1, 1, 6, 12, 20, 18, 23, 18, 14, 5, 1, 1, 6, 15, 23, 28, 28, 29, 24, 16, 6, 1, 1, 7, 17, 30, 34, 38, 37, 40, 29, 19, 6, 1, 1, 7, 20, 35, 46, 52, 51, 52, 50
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OFFSET

0,5


COMMENTS

The top row of the array is where m=0. The leftmost column of the array is where n=1.
Clarification regarding the definition: Each positive integer can be thought of as a finite binary string with 1 as the leftmost digit. The "runs" alternate between those completely of 1's and those completely of 0's. Each run of digit b (0 or 1) is bounded by the digit 1b or by the edge of the string. By "multiset of lengths (in respect to the values of the digits)" of runs, it is meant that the lengths of the runs of digit b's (b=0 or 1) form a permutation of the lengths of the runs of b's in all binary number with the same multisets of the lengths of runs. We are concerned with two multisets, those of the lengths of the runs of 0's, and those of the lengths of the runs of 1's. (See example.)


LINKS

Table of n, a(n) for n=0..86.


EXAMPLE

Consider those binary numbers with exactly four 1's and two 0's. There are 10 such binary numbers that each have a 1 as the leftmost digit. These binary numbers, grouped by those numbers with the same types of runs, are: (111100), (111010, 101110), (111001, 100111), (110110), (110101, 101101, 101011), (110011). There are 6 such groupings, so a(2,4) = 6.


CROSSREFS

Sequence in context: A184441 A172279 A348285 * A136622 A025474 A136575
Adjacent sequences: A164950 A164951 A164952 * A164954 A164955 A164956


KEYWORD

base,nonn,tabl


AUTHOR

Leroy Quet, Sep 01 2009


EXTENSIONS

Definition and comment line improved by Leroy Quet, Sep 02 2009
Extended by Ray Chandler, Mar 14 2010


STATUS

approved



