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

%I A173519 #35 Sep 07 2016 11:08:50
%S A173519 1,1,2,7,23,84,331,1367,5812,25331,112804,511045,2348042,10919414,
%T A173519 51313463,243332340,1163105227,5598774334,27119990519,132107355553,
%U A173519 646793104859,3181256110699,15712610146876,77903855239751,387609232487489,1934788962992123
%N A173519 Number of partitions of n*(n+1)/2 into parts not greater than n.
%C A173519 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 A173519 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 A173519 a(n) = A026820(A000217(n),n).
%F A173519 a(n) ~ c * d^n / n^2, where d = 5.4008719041181541524660911191042700520294... = A258234, c = 0.6326058791290010900659134913629203727... . - _Vaclav Kotesovec_, Sep 07 2014
%t A173519 Table[Length[IntegerPartitions[n(n + 1)/2, n]], {n, 10}] (* _Alonso del Arte_, Aug 12 2011 *)
%t A173519 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 A173519 (PARI)
%o A173519 a(n)=
%o A173519 {
%o A173519     local(tr=n*(n+1)/2, x='x+O('x^(tr+3)), gf);
%o A173519     gf = 1 / prod(k=1,n, 1-x^k); /* g.f. for partitions into parts <=n */
%o A173519     return( polcoeff( truncate(gf), tr ) );
%o A173519 } /* _Joerg Arndt_, Aug 14 2011 */
%Y A173519 Cf. A066655, A097356, A258234.
%K A173519 nonn
%O A173519 0,3
%A A173519 _Reinhard Zumkeller_, Feb 20 2010
%E A173519 More terms from _D. S. McNeil_, Aug 12 2011