login
A173519
Number of partitions of n*(n+1)/2 into parts not greater than n.
11
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
0,3
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)
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
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 *)
PROG
(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 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Feb 20 2010
EXTENSIONS
More terms from D. S. McNeil, Aug 12 2011
STATUS
approved