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%I #18 May 12 2022 17:33:32
%S 1,1,1,1,1,2,2,3,4,5,6,8,10,12,16,19,24,31,37,46,58,70,86,104,127,156,
%T 185,222,273,326,392,463,556,669,792,939,1109,1317,1564,1838,2156,
%U 2535,2986,3514,4100,4777,5577,6526,7621,8847,10251,11869,13807,16032,18529,21370
%N Maximum number of partitions of n into exactly k parts, each <= k. a(n) is maximum in each row of A157044.
%C Without the constraint on each part being <= k: see A008284 and A002569.
%H Alois P. Heinz, <a href="/A157046/b157046.txt">Table of n, a(n) for n = 0..1000</a>
%e For n=9 the counts of partitions for k=1..9 is 0,0,1,4,5,3,2,1,1 so the maximum is 5 (at k=5).
%p b:= proc(n, i, t) option remember; `if`(n=0, 1,
%p `if`(i*t<n, 0, b(n, i-1, t)+b(n-i, min(n-i, i), t-1)))
%p end:
%p a:= n-> max(seq(b(n-i, min(n-i, i-1), i), i=0..n)):
%p seq(a(n), n=0..55); # _Alois P. Heinz_, May 12 2022
%t Max @@@ Table[T[n,k,k]-T[n,k-1,k],{n,1,128},{k,n}] (* with T[n,a,b] as defined in A047993 *)
%Y Cf. A157044, A047993, A008284, A002569.
%K nonn
%O 0,6
%A _Wouter Meeussen_, Feb 22 2009
%E a(0)=1 prepended by _Alois P. Heinz_, May 12 2022
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