%I #18 Jan 20 2020 21:42:10

%S 0,0,1,0,0,1,0,0,1,2,0,0,0,0,2,0,0,0,0,2,1,0,0,0,0,0,1,1,0,0,0,0,0,1,

%T 2,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,0,2,1,0,0,

%U 0,0,0,0,0,0,0,2,1,2,0,0,0,0,0,0,0,0,0,0,1,2,2

%N Square array A(n>=0,k>=0) where A(n,k) gives the (k+1)-th part of the unordered partition which has been encoded in the binary expansion of n, as explained in A227183. The array is scanned antidiagonally as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), etc.

%C Discarding the trailing zero terms, on each row n there is a unique partition of integer A227183(n). All possible partitions of finite natural numbers eventually occur. The first partition that sums to n occurs at row A227368(n).

%C Irregular table A227739 lists only the nonzero terms.

%H Antti Karttunen, <a href="/A227189/b227189.txt">The first 141 antidiagonals of the table, flattened</a>

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%e The top-left corner of the array:

%e row # row starts as

%e 0 0, 0, 0, 0, 0, ...

%e 1 1, 0, 0, 0, 0, ...

%e 2 1, 1, 0, 0, 0, ...

%e 3 2, 0, 0, 0, 0, ...

%e 4 2, 2, 0, 0, 0, ...

%e 5 1, 1, 1, 0, 0, ...

%e 6 1, 2, 0, 0, 0, ...

%e 7 3, 0, 0, 0, 0, ...

%e 8 3, 3, 0, 0, 0, ...

%e 9 1, 2, 2, 0, 0, ...

%e 10 1, 1, 1, 1, 0, ...

%e 11 2, 2, 2, 0, 0, ...

%e 12 2, 3, 0, 0, 0, ...

%e 13 1, 1, 2, 0, 0, ...

%e 14 1, 3, 0, 0, 0, ...

%e 15 4, 0, 0, 0, 0, ...

%e 16 4, 4, 0, 0, 0, ...

%e 17 1, 3, 3, 0, 0, ...

%e etc.

%e 8 has binary expansion "1000", whose runlengths are [3,1] (the length of the run in the least significant end comes first) which maps to nonordered partition {3+3} as explained in A227183, thus row 8 begins as 3, 3, 0, 0, ...

%e 17 has binary expansion "10001", whose runlengths are [1,3,1] which maps to nonordered partition {1,3,3}, thus row 17 begins as 1, 3, 3, ...

%o (Scheme)

%o (define (A227189 n) (A227189bi (A002262 n) (A025581 n)))

%o (define (A227189bi n k) (cond ((< (A005811 n) (+ 1 k)) 0) ((zero? k) (A136480 n)) (else (+ (- (A136480 n) 1) (A227189bi (A163575 n) (- k 1))))))

%Y Only nonzero terms: A227739. Row sums: A227183. The product of nonzero terms on row n>0 is A227184(n). Number of nonzero terms on each row: A005811. The leftmost column, after n>0: A136480. The rightmost nonzero term: A227185.

%Y Cf. A227368 and also arrays A227186 and A227188.

%K nonn,tabl

%O 0,10

%A _Antti Karttunen_, Jul 06 2013