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A238016 Number A(n,k) of partitions of n^k into parts that are at most n; square array A(n,k), n>=0, k>=0, read by antidiagonals. 24

%I

%S 0,1,1,1,1,1,1,1,2,1,1,1,3,3,1,1,1,5,12,5,1,1,1,9,75,64,7,1,1,1,17,

%T 588,2280,377,11,1,1,1,33,5043,123464,106852,2432,15,1,1,1,65,44652,

%U 7566280,55567352,6889527,16475,22,1

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

%C In general, for k>3, is column k asymptotic to exp(2*n) * n^((k-2)*n-k) / (2*Pi). For k=1 see A000041, for k=2 see A206226 and for k=3 see A238608. - _Vaclav Kotesovec_, May 25 2015

%C Conjecture: If f(n) >= O(n^4) then "number of partitions of f(n) into parts that are at most n" is asymptotic to f(n)^(n-1) / (n!*(n-1)!). See also A237998, A238000, A236810 or A258668-A258672. - _Vaclav Kotesovec_, Jun 07 2015

%H Alois P. Heinz, <a href="/A238016/b238016.txt">Antidiagonals n = 0..54, flattened</a>

%H 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 A(n,k) = [x^(n^k)] Product_{j=1..n} 1/(1-x^j).

%e A(3,1) = 3: 3, 21, 111.

%e A(3,2) = 12: 333, 3222, 3321, 22221, 32211, 33111, 222111, 321111, 2211111, 3111111, 21111111, 111111111.

%e A(2,3) = 5: 2222, 22211, 221111, 2111111, 11111111.

%e A(2,4) = 9: 22222222, 222222211, 2222221111, 22222111111, 222211111111, 2221111111111, 22111111111111, 211111111111111, 1111111111111111.

%e Square array A(n,k) begins:

%e 0, 1, 1, 1, 1, 1, ...

%e 1, 1, 1, 1, 1, 1, ...

%e 1, 2, 3, 5, 9, 17, ...

%e 1, 3, 12, 75, 588, 5043, ...

%e 1, 5, 64, 2280, 123464, 7566280, ...

%e 1, 7, 377, 106852, 55567352, 33432635477, ...

%t A[n_, k_] := SeriesCoefficient[Product[1/(1-x^j), {j, 1, n}], {x, 0, n^k}]; A[0, 0] = 0; Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* _Jean-Fran├žois Alcover_, Oct 11 2015 *)

%Y Columns k=0-10 give: A057427, A000041, A206226, A238608, A238609, A238610, A238611, A238612, A238613, A238614, A238615.

%Y Rows n=0-10 give: A057427, A000012, A094373, A238630, A238631, A238632, A238633, A238634, A238635, A238636, A238637.

%Y Main diagonal gives A238000.

%Y Cf. A238010.

%K nonn,tabl

%O 0,9

%A _Alois P. Heinz_, Feb 17 2014

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Last modified July 31 01:15 EDT 2021. Contains 346365 sequences. (Running on oeis4.)