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A227739
Irregular table where row n lists in nondecreasing order the parts of unordered partition encoded in the runlengths of binary expansion of n; nonzero terms of A227189.
15
1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 3, 3, 3, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 3, 1, 1, 2, 1, 3, 4, 4, 4, 1, 3, 3, 1, 1, 2, 2, 2, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 1, 2, 3, 1, 1, 1, 2, 2, 2, 3, 2, 4, 1, 1, 3, 1, 4, 5, 5, 5, 1, 4, 4, 1, 1
OFFSET
1,4
COMMENTS
Row n has A005811(n) elements. Each row contains a unique (unordered) partition of some integer, and all possible partitions of finite natural numbers eventually occur. The first partition that sums to k occurs at row A227368(k) and the last at row A000225(k).
Other similar tables of unordered partitions: A036036, A036037, A080576, A080577 and A112798.
FORMULA
a(n) = A227189(A227737(n),A227740(n)).
EXAMPLE
Rows are constructed as:
Row n in Runlengths With one Partial sums The row sums
n binary collected subtracted of which give to, i.e. is
from lsb- from all terms on a partition of
to msb-end except 1st that row of A227183(n)
1 "1" [1] [1] 1; 1
2 "10" [1,1] [1,0] 1, 1; 2
3 "11" [2] [2] 2; 2
4 "100" [2,1] [2,0] 2, 2; 4
5 "101" [1,1,1] [1,0,0] 1, 1, 1; 3
6 "110" [1,2] [1,1] 1, 2; 3
7 "111" [3] [3] 3; 3
8 "1000" [3,1] [3,0] 3, 3; 6
9 "1001" [1,2,1] [1,1,0] 1, 2, 2; 5
10 "1010" [1,1,1,1] [1,0,0,0] 1, 1, 1, 1; 4
11 "1011" [2,1,1] [2,0,0] 2, 2, 2; 6
12 "1100" [2,2] [2,1] 2, 3; 5
13 "1101" [1,1,2] [1,0,1] 1, 1, 2; 4
14 "1110" [1,3] [1,2] 1, 3; 4
15 "1111" [4] [4] 4; 4
16 "10000" [4,1] [4,0] 4, 4; 8
MATHEMATICA
Table[Function[b, Accumulate@ Prepend[If[Length@ b > 1, Rest[b] - 1, {}], First@ b]]@ Map[Length, Split@ Reverse@ IntegerDigits[n, 2]], {n, 34}] // Flatten (* Michael De Vlieger, May 09 2017 *)
PROG
(Scheme) (define (A227739 n) (A227189bi (A227737 n) (A227740 n))) ;; The Scheme-code for A227189bi has been given in A227189.
CROSSREFS
Row sums: A227183, row products: A227184, the initial (smallest) term of each row: A136480, the last (largest) term: A227185.
Cf. also A227189, A227738, A227736.
Sequence in context: A351206 A176508 A241492 * A047971 A029432 A073426
KEYWORD
nonn,base,tabf
AUTHOR
Antti Karttunen, Jul 25 2013
STATUS
approved