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 A227189 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. 9
 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, 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, 0, 0, 0, 0, 0, 0, 0, 2, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,10 COMMENTS Discarding the trailing zero terms, on each row n there is an 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). Irregular table A227739 lists only the nonzero terms. LINKS Antti Karttunen, The first 141 antidiagonals of the table, flattened EXAMPLE The top-left corner of the array: row #  row starts as     0  0, 0, 0, 0, 0, ...     1  1, 0, 0, 0, 0, ...     2  1, 1, 0, 0, 0, ...     3  2, 0, 0, 0, 0, ...     4  2, 2, 0, 0, 0, ...     5  1, 1, 1, 0, 0, ...     6  1, 2, 0, 0, 0, ...     7  3, 0, 0, 0, 0, ...     8  3, 3, 0, 0, 0, ...     9  1, 2, 2, 0, 0, ...    10  1, 1, 1, 1, 0, ...    11  2, 2, 2, 0, 0, ...    12  2, 3, 0, 0, 0, ...    13  1, 1, 2, 0, 0, ...    14  1, 3, 0, 0, 0, ...    15  4, 0, 0, 0, 0, ...    16  4, 4, 0, 0, 0, ...    17  1, 3, 3, 0, 0, ... etc. 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, ... 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, ... PROG (Scheme) (define (A227189 n) (A227189bi (A002262 n) (A025581 n))) (define (A227189bi n k) (cond ((< (A005811 n) (+ 1 k)) 0) ((zero? k) (A136480 n)) (else (+ (- (A136480 n) 1) (A227189bi (A163575 n) (- k 1)))))) CROSSREFS 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. Cf. A227368 and also arrays A227186 and A227188. Sequence in context: A227186 A037134 A254612 * A234859 A001343 A022882 Adjacent sequences:  A227186 A227187 A227188 * A227190 A227191 A227192 KEYWORD nonn,tabl AUTHOR Antti Karttunen, Jul 06 2013 STATUS approved

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