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A238012 Number A(n,k) of partitions of k^n into parts that are at most n with at least one part of each size; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A238012 #30 Aug 29 2018 21:14:53
%S A238012 0,0,0,0,1,0,0,1,0,0,0,1,1,0,0,0,1,4,2,0,0,0,1,7,48,9,0,0,0,1,12,310,
%T A238012 3042,119,0,0,0,1,17,1240,109809,1067474,4935,0,0,0,1,24,3781,1655004,
%U A238012 370702459,2215932130,596763,0,0,0,1,31,9633,14942231,32796849930,13173778523786,29012104252380,211517867,0,0
%N A238012 Number A(n,k) of partitions of k^n into parts that are at most n with at least one part of each size; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A238012 In general, column k>=2 is asymptotic to k^(n*(n-1)) / (n!*(n-1)!). - _Vaclav Kotesovec_, Jun 05 2015
%H A238012 Alois P. Heinz, <a href="/A238012/b238012.txt">Antidiagonals n = 0..43, flattened</a>
%H A238012 A. V. Sills and D. Zeilberger, <a href="https://arxiv.org/abs/1108.4391">Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz)</a>, arXiv:1108.4391 [math.CO], 2011.
%F A238012 A(n,k) = [x^(k^n-n*(n+1)/2)] Product_{j=1..n} 1/(1-x^j).
%e A238012 Square array A(n,k) begins:
%e A238012   0, 0,   0,       0,         0,           0, ...
%e A238012   0, 1,   1,       1,         1,           1, ...
%e A238012   0, 0,   1,       4,         7,          12, ...
%e A238012   0, 0,   2,      48,       310,        1240, ...
%e A238012   0, 0,   9,    3042,    109809,     1655004, ...
%e A238012   0, 0, 119, 1067474, 370702459, 32796849930, ...
%t A238012 A[0, 0] = 0;
%t A238012 A[n_, k_] := SeriesCoefficient[Product[1/(1-x^j), {j, 1, n}], {x, 0, k^n - n(n+1)/2}];
%t A238012 Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Aug 18 2018, after _Alois P. Heinz_ *)
%Y A238012 Columns k=0-10 give: A000004, A063524, A237999, A239162, A239163, A239164, A239165, A239166, A239167, A239168, A239169.
%Y A238012 Rows n=0-2 give: A000004, A057427, A074148(k-1) for k>1.
%Y A238012 Main diagonal gives A238001.
%Y A238012 Cf. A238010.
%K A238012 nonn,tabl
%O A238012 0,18
%A A238012 _Alois P. Heinz_, Feb 16 2014