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A144512 Array read by upwards antidiagonals: T(n,k) = total number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1, 2, ..., k+1, for 0 <= k <= (k+1)*n.

%I A144512 #12 Feb 01 2017 18:13:38
%S A144512 1,1,1,1,2,1,1,3,7,1,1,4,31,37,1,1,5,121,842,266,1,1,6,456,18252,
%T A144512 45296,2431,1,1,7,1709,405408,7958726,4061871,27007,1,1,8,6427,
%U A144512 9268549,1495388159,7528988476,546809243,353522,1,1,9,24301,216864652,295887993624,15467641899285
%N A144512 Array read by upwards antidiagonals: T(n,k) = total number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1, 2, ..., k+1, for 0 <= k <= (k+1)*n.
%H A144512 Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, <a href="http://arxiv.org/abs/1701.08394">Analysis of the Gift Exchange Problem</a>, arXiv:1701.08394, 2017.
%H A144512 David Applegate and N. J. A. Sloane, <a href="http://arxiv.org/abs/0907.0513">The Gift Exchange Problem</a> (arXiv:0907.0513, 2009)
%e A144512 Array begins:
%e A144512 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e A144512 1, 2, 7, 37, 266, 2431, 27007, 353522, 5329837, ...
%e A144512 1, 3, 31, 842, 45296, 4061871, 546809243, 103123135501, ...
%e A144512 1, 4, 121, 18252, 7958726, 7528988476, 13130817809439, ...
%e A144512 1, 5, 456, 405408, 1495388159, 15467641899285, 361207016885536095, ...
%e A144512 1, 6, 1709, 9268549, 295887993624, 34155922905682979, 10893033763705794846727, ...
%e A144512 ...
%p A144512 b := proc(n, i, k) local r;
%p A144512 option remember;
%p A144512 if n = i then 1;
%p A144512 elif i < n then 0;
%p A144512 elif n < 1 then 0;
%p A144512 else add( binomial(i-1,r)*b(n-1,i-1-r,k), r=0..k);
%p A144512 end if;
%p A144512 end proc;
%p A144512 T:=proc(n,k); add(b(n,i,k),i=0..(k+1)*n); end proc;
%t A144512 multinomial[n_, k_List] := n!/Times @@ (k!); t[n_, k_] := Module[{i, ik}, ik = Array[i, k]; 1/k!* Sum[multinomial[Total[ik], ik], Evaluate[Sequence @@ Thread[{ik, 1, n}]]]]; Table[t[n-k, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* _Jean-François Alcover_, Jan 14 2014 *)
%Y A144512 See A144510 for Maple code.
%Y A144512 Rows include A001515, A144416, A144508, A144509, A149187.
%Y A144512 Columns include A048775, A144511, A144662, A147984.
%Y A144512 Transpose of array in A144510.
%Y A144512 Main diagonal gives A281901.
%K A144512 nonn,tabl
%O A144512 0,5
%A A144512 _David Applegate_ and _N. J. A. Sloane_, Dec 15 2008, Dec 21 2008