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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
 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, 588, 2280, 377, 11, 1, 1, 1, 33, 5043, 123464, 106852, 2432, 15, 1, 1, 1, 65, 44652, 7566280, 55567352, 6889527, 16475, 22, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 COMMENTS 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 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 LINKS Alois P. Heinz, Antidiagonals n = 0..54, flattened A. V. Sills and D. Zeilberger, Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz) arXiv:1108.4391 [math.CO], 2011. FORMULA A(n,k) = [x^(n^k)] Product_{j=1..n} 1/(1-x^j). EXAMPLE A(3,1) = 3: 3, 21, 111. A(3,2) = 12: 333, 3222, 3321, 22221, 32211, 33111, 222111, 321111, 2211111, 3111111, 21111111, 111111111. A(2,3) = 5: 2222, 22211, 221111, 2111111, 11111111. A(2,4) = 9: 22222222, 222222211, 2222221111, 22222111111, 222211111111, 2221111111111, 22111111111111, 211111111111111, 1111111111111111. Square array A(n,k) begins:   0, 1,   1,      1,        1,           1, ...   1, 1,   1,      1,        1,           1, ...   1, 2,   3,      5,        9,          17, ...   1, 3,  12,     75,      588,        5043, ...   1, 5,  64,   2280,   123464,     7566280, ...   1, 7, 377, 106852, 55567352, 33432635477, ... MATHEMATICA 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-François Alcover, Oct 11 2015 *) CROSSREFS Columns k=0-10 give: A057427, A000041, A206226, A238608, A238609, A238610, A238611, A238612, A238613, A238614, A238615. Rows n=0-10 give: A057427, A000012, A094373, A238630, A238631, A238632, A238633, A238634, A238635, A238636, A238637. Main diagonal gives A238000. Cf. A238010. Sequence in context: A220708 A110541 A331461 * A185812 A152798 A079115 Adjacent sequences:  A238013 A238014 A238015 * A238017 A238018 A238019 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Feb 17 2014 STATUS approved

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Last modified June 14 08:09 EDT 2021. Contains 345018 sequences. (Running on oeis4.)