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%I #19 Jan 12 2024 16:33:25
%S 1,0,2,0,1,2,0,1,2,2,0,0,3,2,2,0,0,3,4,2,2,0,0,2,5,4,2,2,0,0,1,7,6,4,
%T 2,2,0,0,1,6,9,6,4,2,2,0,0,0,7,11,10,6,4,2,2,0,0,0,5,14,13,10,6,4,2,2,
%U 0,0,0,5,15,19,14,10,6,4,2,2,0,0,0,3,17,22,21,14,10,6,4,2,2,0,0,0,2,17,29
%N Triangle read by rows: T(n,m) is the number of partitions of n that (just) fit inside an m X m box, but not in an (m-1) X (m-1) box. Partitions of n with Max(max part, length) = m.
%C Row sums are A000041. Columns are finite and sum to A051924. The final floor(n/2) terms of each row are the reverse of the initial terms of 2*A000041.
%H Andrew Howroyd, <a href="/A096771/b096771.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%H Findstat, <a href="http://www.findstat.org/StatisticsDatabase/St000784">FindStat - St000784: The maximum of the length and the largest part of the integer partition</a>
%F Sum_{k>=1} k*T(n,k) = A368985(n). - _Andrew Howroyd_, Jan 12 2024
%e T(5,3)=3, counting 32, 311 and 221.
%e From _Gus Wiseman_, Apr 12 2019: (Start)
%e Triangle begins:
%e 1
%e 0 2
%e 0 1 2
%e 0 1 2 2
%e 0 0 3 2 2
%e 0 0 3 4 2 2
%e 0 0 2 5 4 2 2
%e 0 0 1 7 6 4 2 2
%e 0 0 1 6 9 6 4 2 2
%e 0 0 0 7 11 10 6 4 2 2
%e 0 0 0 5 14 13 10 6 4 2 2
%e 0 0 0 5 15 19 14 10 6 4 2 2
%e 0 0 0 3 17 22 21 14 10 6 4 2 2
%e 0 0 0 2 17 29 27 22 14 10 6 4 2 2
%e 0 0 0 1 17 33 36 29 22 14 10 6 4 2 2
%e 0 0 0 1 15 39 45 41 30 22 14 10 6 4 2 2
%e 0 0 0 0 14 41 57 52 43 30 22 14 10 6 4 2 2
%e 0 0 0 0 11 47 67 69 57 44 30 22 14 10 6 4 2 2
%e 0 0 0 0 9 46 81 85 76 59 44 30 22 14 10 6 4 2 2
%e (End)
%t Table[Count[Partitions[n], q_List /; Max[Length[q], Max[q]]===k], {n, 16}, {k, n}]
%o (PARI) row(n)={my(r=vector(n)); forpart(p=n, r[max(#p,p[#p])]++); r} \\ _Andrew Howroyd_, Jan 12 2024
%Y A version with reflected rows is A338621.
%Y Cf. A051924, A096272, A096597, A115994, A252464, A263297, A325193, A368985.
%Y Related triangles are A115720, A325188, A325189, A325192, A325200, with Heinz-encoded versions A257990, A325169, A065770, A325178, A325195.
%K easy,nonn,tabl
%O 1,3
%A _Wouter Meeussen_, Aug 21 2004
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