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A238010 Number A(n,k) of partitions of k^n into parts that are at most n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A238010 #31 Aug 29 2018 21:14:03
%S A238010 0,0,1,0,1,1,0,1,1,1,0,1,3,1,1,0,1,5,10,1,1,0,1,9,75,64,1,1,0,1,13,
%T A238010 374,4410,831,1,1,0,1,19,1365,123464,1366617,26207,1,1,0,1,25,3997,
%U A238010 1736385,393073019,2559274110,2239706,1,1
%N A238010 Number A(n,k) of partitions of k^n into parts that are at most n; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A238010 In general, column k>=2 is asymptotic to k^(n*(n-1)) / (n!*(n-1)!). - _Vaclav Kotesovec_, Jun 05 2015
%H A238010 Alois P. Heinz, <a href="/A238010/b238010.txt">Antidiagonals n = 0..43, flattened</a>
%H A238010 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])
%F A238010 A(n,k) = [x^(k^n)] Product_{j=1..n} 1/(1-x^j).
%e A238010 A(3,2) = 10: 332, 2222, 3221, 3311, 22211, 32111, 221111, 311111, 2111111, 11111111.
%e A238010 A(2,3) = 5: 22221, 222111, 2211111, 21111111, 111111111.
%e A238010 A(2,4) = 9: 22222222, 222222211, 2222221111, 22222111111, 222211111111, 2221111111111, 22111111111111, 211111111111111, 1111111111111111.
%e A238010 Square array A(n,k) begins:
%e A238010   0, 0,   0,       0,         0,           0, ...
%e A238010   1, 1,   1,       1,         1,           1, ...
%e A238010   1, 1,   3,       5,         9,          13, ...
%e A238010   1, 1,  10,      75,       374,        1365, ...
%e A238010   1, 1,  64,    4410,    123464,     1736385, ...
%e A238010   1, 1, 831, 1366617, 393073019, 33432635477, ...
%t A238010 A[n_, k_] := SeriesCoefficient[Product[1/(1-x^j), {j, 1, n}], {x, 0, k^n}]; A[0, 0] = 0; Table[A[n-k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Feb 17 2017 *)
%Y A238010 Columns k=0+1,2-10 give: A057427, A237998, A238560, A238561, A238562, A238563, A238564, A238565, A238566, A238567.
%Y A238010 Rows n=0-2 give: A000004, A000012, A080827.
%Y A238010 Main diagonal gives A238000.
%Y A238010 Cf. A238012, A238016.
%K A238010 nonn,tabl
%O A238010 0,13
%A A238010 _Alois P. Heinz_, Feb 16 2014