A173519 Number of partitions of n*(n+1)/2 into parts not greater than n.

1, 1, 2, 7, 23, 84, 331, 1367, 5812, 25331, 112804, 511045, 2348042, 10919414, 51313463, 243332340, 1163105227, 5598774334, 27119990519, 132107355553, 646793104859, 3181256110699, 15712610146876, 77903855239751, 387609232487489, 1934788962992123

Offset: 0

Author

Reinhard Zumkeller, Feb 20 2010

Keywords

• nonn

Comments

a(n) is also the number of partitions of n^3 into n distinct parts <= n*(n+1). a(3) = 7: [4,11,12], [5,10,12], [6,9,12], [6,10,11], [7,8,12], [7,9,11], [8,9,10]. - Alois P. Heinz, Jan 25 2012

Links

• Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..720 (terms 0..200 from Alois P. Heinz)

Crossrefs

Cf. A066655, A097356, A258234.

Programs

Mathematica

Table[Length[IntegerPartitions[n(n + 1)/2, n]], {n, 10}] (* Alonso del Arte, Aug 12 2011 *)
Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,n*(n+1)/2}],{n,0,20}] (* Vaclav Kotesovec, May 25 2015 *)

PARI

a(n)=
{
    local(tr=n*(n+1)/2, x='x+O('x^(tr+3)), gf);
    gf = 1 / prod(k=1,n, 1-x^k); /* g.f. for partitions into parts <=n */
    return( polcoeff( truncate(gf), tr ) );
} /* Joerg Arndt, Aug 14 2011 */

Formula

a(n) = A026820(A000217(n),n).

a(n) ~ c * d^n / n^2, where d = 5.4008719041181541524660911191042700520294... = A258234, c = 0.6326058791290010900659134913629203727... . - Vaclav Kotesovec, Sep 07 2014

Extensions

More terms from D. S. McNeil, Aug 12 2011