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%I #27 Jan 14 2019 03:06:35
%S 1,0,1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,1,1,0,0,0,2,1,1,1,0,0,0,1,2,1,
%T 1,1,0,0,0,1,2,2,1,1,1,0,0,0,1,2,2,2,1,1,1,0,0,0,0,2,3,2,2,1,1,1,0,0,
%U 0,0,2,3,3,2,2,1,1,1,0,0,0,0,1,3,4,3,2,2,1,1
%N Triangular array T read by rows: T(n,k) = number of partitions of n into distinct parts, the greatest being k, for k=1,2,...,n.
%C Conjecture: A199918(n) = Sum_{k=1..n} (-1)^(n-k) T(n,k). - _George Beck_, Jan 13 2019
%H Henry Bottomley, <a href="http://www.se16.info/js/partitions.htm">Partition calculators using java applets</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F T(n, k) = A070936(n-k, k-1) = A053632(k-1, n-k) = T(n-1, k-1)+T(n-2k+1, k-1). - _Henry Bottomley_, May 12 2002
%F T(n, k) = coefficient of x^n in x^k*Product_{i=1..k-1} (1+x^i). - _Vladeta Jovovic_, Aug 07 2003
%e Triangle begins:
%e [1]
%e [0, 1]
%e [0, 1, 1]
%e [0, 0, 1, 1]
%e [0, 0, 1, 1, 1]
%e [0, 0, 1, 1, 1, 1]
%e [0, 0, 0, 2, 1, 1, 1]
%e [0, 0, 0, 1, 2, 1, 1, 1]
%e [0, 0, 0, 1, 2, 2, 1, 1, 1]
%e [0, 0, 0, 1, 2, 2, 2, 1, 1, 1]
%e [0, 0, 0, 0, 2, 3, 2, 2, 1, 1, 1]
%e [0, 0, 0, 0, 2, 3, 3, 2, 2, 1, 1, 1]
%e [0, 0, 0, 0, 1, 3, 4, 3, 2, 2, 1, 1, 1]
%e [0, 0, 0, 0, 1, 3, 4, 4, 3, 2, 2, 1, 1, 1]
%e ... - _N. J. A. Sloane_, Nov 09 2018
%p with(combinat);
%p f2:=proc(n) local i,j,p,t0,t1,t2;
%p t0:=Array(1..n,0);
%p t1:=partition(n);
%p p:=numbpart(n);
%p for i from 1 to p do
%p t2:=t1[i];
%p if nops(convert(t2,set))=nops(t2) then
%p # now have a partition t2 of n into distinct parts
%p t0[t2[-1]]:=t0[t2[-1]]+1;
%p od:
%p [seq(t0[j],j=1..n)];
%p end proc;
%p for n from 1 to 12 do lprint(f2(n)); od: # _N. J. A. Sloane_, Nov 09 2018
%Y If seen as a square array then transpose of A070936 and visible form of A053632. Central diagonal and those to the right of center are A000009 as are row sums.
%K nonn,tabl
%O 1,25
%A _Clark Kimberling_
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