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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 (list; table; graph; refs; listen; history; text; internal format)
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 1-b 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: A006375 A184441 A172279 * 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

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Last modified June 28 16:56 EDT 2017. Contains 288839 sequences.