A206226 Number of partitions of n^2 into parts not greater than n.

1, 1, 3, 12, 64, 377, 2432, 16475, 116263, 845105, 6292069, 47759392, 368379006, 2879998966, 22777018771, 181938716422, 1465972415692, 11902724768574, 97299665768397, 800212617435074, 6617003142869419, 54985826573015541, 458962108485797208, 3846526994743330075

Offset: 0

Author

Paul D. Hanna, Feb 05 2012

Keywords

• nonn

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)

Crossrefs

Cf. A173519, A206227, A206240, A107379, A258268.

Column k=2 of A238016.

Cf. A258296 (j=2), A258293 (j=3), A258294 (j=4), A258295 (j=5).

Programs

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 *)

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),", "))

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