OFFSET
0,3
COMMENTS
Also the number of partitions of n^2 using n or fewer numbers. Thus for n=3 one has: 9; 1,8; 2,7; 3,6; 4,5; 1,1,7; 1,2,6; 1,3,5; 1,4,4; 2,2,5; 2,3,4; 3,3,3. - J. M. Bergot, Mar 26 2014 [computations done by Charles R Greathouse IV]
The partitions in the comments above are the conjugates of the partitions in the definition. By conjugation we have: "partitions into parts <= m" are equinumerous with "partitions into at most m parts". - Joerg Arndt, Mar 31 2014
From Vaclav Kotesovec, May 25 2015: (Start)
In general, "number of partitions of j*n^2 into parts that are at most n" is (for j>0) asymptotic to c(j) * d(j)^n / n^2, where c(j) and d(j) are a constants.
-------
j c(j)
1 0.1582087202672504149766310999238...
2 0.0794245035465730707705885572860...
3 0.0530017980244665552354063060738...
4 0.0397666338404544208556554596295...
5 0.0318193213988281353709268311928...
...
17 0.0093617308583114626385718275875...
c(j) for big j asymptotically approaches 1 / (2*Pi*j).
---------
j d(j)
1 9.15337019245412246194853029240... = A258268
2 16.57962120993269533568313969522...
3 23.98280768122086592445663786762...
4 31.37931997386325137074644287711...
5 38.77298550971449870728474612568...
...
17 127.45526806942537991146993713837...
d(j) for big j asymptotically approaches j * exp(2).
(End)
d(j) = r^(2*j+1)/(r-1), where r is the root of the equation polylog(2, 1-r) + (j+1/2)*log(r)^2 = 0. - Vaclav Kotesovec, Jun 11 2015
LINKS
Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..382 (first 150 terms from Alois P. Heinz)
FORMULA
a(n) = [x^(n^2)] Product_{k=1..n} 1/(1 - x^k).
a(n) ~ c * d^n / n^2, where d = 9.1533701924541224619485302924013545... = A258268, c = 0.1582087202672504149766310999238742... . - Vaclav Kotesovec, Sep 07 2014
MAPLE
T:= proc(n, k) option remember;
`if`(n=0 or k=1, 1, T(n, k-1) + `if`(k>n, 0, T(n-k, k)))
end:
seq(T(n^2, n), n=0..20); # Vaclav Kotesovec, May 25 2015 after Alois P. Heinz
MATHEMATICA
Table[SeriesCoefficient[Product[1/(1-x^k), {k, 1, n}], {x, 0, n^2}], {n, 0, 20}] (* Vaclav Kotesovec, May 25 2015 *)
(* A program to compute the constants d(j) *) Table[r^(2*j+1)/(r-1) /.FindRoot[-PolyLog[2, 1-r] == (j+1/2)*Log[r]^2, {r, E}, WorkingPrecision->60], {j, 1, 5}] (* Vaclav Kotesovec, Jun 11 2015 *)
PROG
(PARI) {a(n)=polcoeff(prod(k=1, n, 1/(1-x^k+x*O(x^(n^2)))), n^2)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 05 2012
STATUS
approved