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%I #12 Nov 09 2018 20:33:59
%S 1,0,1,0,0,1,0,0,1,2,0,0,0,3,3,0,0,0,3,17,6,0,0,0,1,36,74,11,0,0,0,1,
%T 60,573,358,23,0,0,0,0,56,2802,7311,1631,47,0,0,0,0,50,10087,107938,
%U 83170,7563,106,0,0,0,0,27,26512,1186969,3121840,866657,34751,235,0,0,0,0,19
%N Triangle read by rows: Distinct classifications of N elements containing exactly R binary partitions.
%C Significance of triangle suggested by _Franklin T. Adams-Watters_ on Dec 19 2009. Row N has N terms in this sequence. The triangle starts:
%C 1;
%C 0, 1;
%C 0, 0, 1;
%C 0, 0, 1, 2;
%C 0, 0, 0, 3, 3;
%C 0, 0, 0, 0, 3, 17, 6;
%C 0, 0, 0, 0, 1, 36, 74, 11;
%C Value is A000055(N) when R=N-1 (last term in each row). (Conjectured by _Robert Munafo_ Dec 28 2009, then proved by _Andrew Weimholt_ and _Franklin T. Adams-Watters_ on Dec 29 2009)
%C Value is 1 when N=2^R.
%C Value is 1 when N=(2^R)-1.
%C Value is R when R>2 and N=(2^R)-2.
%C Value is A034198(R) when R>2 and N=(2^R)-3.
%C Conjecture: In general, in each column, the last 2^(R-1) values are the same as the first 2^(N-1) values from the corresponding row of A039754. - _Robert Munafo_, Dec 30 2009
%C Value is 0 for all (N,R) for which N is greater than 2^R.
%C Each term A(N,R) can be computed most efficiently by first enumerating all classifications in A(N-1,R) plus those in A(N-1,R-1), and then adding an additional type and/or partition to each.
%H R. Munafo, <a href="http://mrob.com/pub/math/seq-a005646.html">Classifications of N Elements</a>
%Y Cf. Row sums are A005646, column sums are A171832.
%Y Cf. A039754.
%Y Last term in each row is A000055(N).
%Y Same triangle read by columns is A171872.
%K nonn,tabl
%O 0,10
%A _Robert Munafo_, Jan 21 2010
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