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%I #27 Nov 17 2022 05:24:53
%S 1,1,2,2,1,3,2,3,3,1,4,3,3,4,3,5,1,5,3,6,2,1,6,4,7,2,2,1,7,6,7,3,2,2,
%T 1,8,8,8,3,2,3,3,8,9,11,3,2,3,3,3,8,10,15,3,2,3,3,4,1,0,1,10,11,18,4,
%U 2,3,3,5,1,1,1,0,0,0,1,14,12,20,5,3,3,3,5,1,2,2,0,0,0,1,0,0,1,17,14,23,5,5
%N Table with T(1,1) = 1; for n>1, T(n,k) is the number of k's in rows 1 through n-1.
%C Equivalently, list 1, where, at stage k>1, write i in list 1 and j in list 2, where i is the number of j's in list 1, for j=1,2,...,m, where m=max number in list 1 from stages 1 to k-1; stage 1 is 1 in list 1.
%C Differs from A030717 in that this sequence includes 0's. - _Sean A. Irvine_, Apr 08 2020
%C Nevertheless, this sequence starts each row with the count of 1's, not 0's, whose counts are not recorded in the sequence (cf. A174382, which is also initialized with a 0). - _Peter Munn_, Oct 11 2022
%H Franklin T. Adams-Watters, <a href="/A333867/b333867.txt">First 32 rows, flattened</a>
%H Michael De Vlieger, <a href="/A333867/a333867.png">Plot a(w(j) + k - 1) at (j,k)</a> for j = 1..60 and w the sequence of partial sums of A126027, showing a(m) = 0 in black, a(m) = 1 in red, and a(m) > 1 in light blue.
%H Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a333/A333867.java">Java program</a> (github)
%e 1;
%e 1;
%e 2;
%e 2, 1;
%e 3, 2;
%e 3, 3, 1;
%e 4, 3, 3;
%t t = {{1}}; Do[AppendTo[t, BinCounts[#, {1, Max[#] + 1}] &[Flatten[t]]], {30}];
%t Map[Length, t] (* A126027*)
%t Map[Total, t] (* A253170*)
%t Flatten[t] (* A333867 *) (* _Peter J. C. Moses_, Apr 09 2020 *)
%o (Haskell)
%o import Data.List (sort, group)
%o a030717 n k = a030717_tabf !! (n-1) !! (k-1)
%o a030717_row n = a030717_tabf !! (n-1)
%o a030717_tabf = [1] : f [1] where
%o f xs = ys : f ((filter (> 0) ys) ++ xs) where
%o ys = h (group $ sort xs) [1..] where
%o h [] _ = []
%o h vss'@(vs:vss) (w:ws)
%o | head vs == w = (length vs) : h vss ws
%o | otherwise = 0 : h vss' ws
%o -- _Reinhard Zumkeller_, Dec 28 2014
%Y Cf. A126027 (row lengths), A006920, A030717 (zeros suppressed).
%Y Cf. A174382.
%Y Cf. A253170 (row sums).
%K nonn,tabf,nice
%O 1,3
%A _Clark Kimberling_
%E More terms from _Franklin T. Adams-Watters_, Dec 14 2006
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