A173519 - OEIS

%I #35 Sep 07 2016 11:08:50

%S 1,1,2,7,23,84,331,1367,5812,25331,112804,511045,2348042,10919414,

%T 51313463,243332340,1163105227,5598774334,27119990519,132107355553,

%U 646793104859,3181256110699,15712610146876,77903855239751,387609232487489,1934788962992123

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

%C 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

%H Alois P. Heinz and Vaclav Kotesovec, <a href="/A173519/b173519.txt">Table of n, a(n) for n = 0..720</a> (terms 0..200 from Alois P. Heinz)

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

%F a(n) ~ c * d^n / n^2, where d = 5.4008719041181541524660911191042700520294... = A258234, c = 0.6326058791290010900659134913629203727... . - _Vaclav Kotesovec_, Sep 07 2014

%t Table[Length[IntegerPartitions[n(n + 1)/2, n]], {n, 10}] (* _Alonso del Arte_, Aug 12 2011 *)

%t Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,n*(n+1)/2}],{n,0,20}] (* _Vaclav Kotesovec_, May 25 2015 *)

%o (PARI)

%o a(n)=

%o {

%o     local(tr=n*(n+1)/2, x='x+O('x^(tr+3)), gf);

%o     gf = 1 / prod(k=1,n, 1-x^k); /* g.f. for partitions into parts <=n */

%o     return( polcoeff( truncate(gf), tr ) );

%o } /* _Joerg Arndt_, Aug 14 2011 */

%Y Cf. A066655, A097356, A258234.

%K nonn

%O 0,3

%A _Reinhard Zumkeller_, Feb 20 2010

%E More terms from _D. S. McNeil_, Aug 12 2011