This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A227189 #18 Jan 20 2020 21:42:10
%S A227189 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 A227189 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 A227189 0,0,0,0,0,0,0,2,1,2,0,0,0,0,0,0,0,0,0,0,1,2,2
%N 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.
%C A227189 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 A227189 Irregular table A227739 lists only the nonzero terms.
%H A227189 Antti Karttunen, <a href="/A227189/b227189.txt">The first 141 antidiagonals of the table, flattened</a>
%H A227189 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%e A227189 The top-left corner of the array:
%e A227189 row # row starts as
%e A227189 0 0, 0, 0, 0, 0, ...
%e A227189 1 1, 0, 0, 0, 0, ...
%e A227189 2 1, 1, 0, 0, 0, ...
%e A227189 3 2, 0, 0, 0, 0, ...
%e A227189 4 2, 2, 0, 0, 0, ...
%e A227189 5 1, 1, 1, 0, 0, ...
%e A227189 6 1, 2, 0, 0, 0, ...
%e A227189 7 3, 0, 0, 0, 0, ...
%e A227189 8 3, 3, 0, 0, 0, ...
%e A227189 9 1, 2, 2, 0, 0, ...
%e A227189 10 1, 1, 1, 1, 0, ...
%e A227189 11 2, 2, 2, 0, 0, ...
%e A227189 12 2, 3, 0, 0, 0, ...
%e A227189 13 1, 1, 2, 0, 0, ...
%e A227189 14 1, 3, 0, 0, 0, ...
%e A227189 15 4, 0, 0, 0, 0, ...
%e A227189 16 4, 4, 0, 0, 0, ...
%e A227189 17 1, 3, 3, 0, 0, ...
%e A227189 etc.
%e A227189 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 A227189 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 A227189 (Scheme)
%o A227189 (define (A227189 n) (A227189bi (A002262 n) (A025581 n)))
%o A227189 (define (A227189bi n k) (cond ((< (A005811 n) (+ 1 k)) 0) ((zero? k) (A136480 n)) (else (+ (- (A136480 n) 1) (A227189bi (A163575 n) (- k 1))))))
%Y A227189 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 A227189 Cf. A227368 and also arrays A227186 and A227188.
%K A227189 nonn,tabl
%O A227189 0,10
%A A227189 _Antti Karttunen_, Jul 06 2013