A238608 Number of partitions of n^3 into parts that are at most n.

1, 1, 5, 75, 2280, 106852, 6889527, 569704489, 57733506640, 6944433285769, 968356321790171, 153738253618009045, 27396489338187214000, 5417302365503826145732, 1177436831956414016252071, 279074576444362385794783853, 71649589941044468875380333533

Offset: 0

Author

Alois P. Heinz, Mar 01 2014

Keywords

• nonn

Comments

In general, "number of partitions of j*n^3 into parts that are at most n" is (for j>0) asymptotic to exp(2*n + 1/(4*j)) * n^(n-3) * j^(n-1) / (2*Pi). - Vaclav Kotesovec, May 25 2015

Links

• Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..122 (terms 0..70 from Alois P. Heinz)

Crossrefs

Column k=3 of A238016.

Cf. A258302 (j=2), A258303 (j=3), A258304 (j=4), A258305 (j=5).

Programs

Maple

T:=proc(n,k) option remember; `if`(n=0 or k=1, 1, T(n,k-1) + `if`(nVaclav Kotesovec, May 25 2015 after Alois P. Heinz

Mathematica

a[n_] := SeriesCoefficient[1/QPochhammer[q, q, n], {q, 0, n^3}]; Table[ a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 03 2015 *)

Formula

a(n) = [x^(n^3)] Product_{j=1..n} 1/(1-x^j).

a(n) ~ exp(2*n + 1/4) * n^(n-3) / (2*Pi). - Vaclav Kotesovec, May 25 2015