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A173519
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Number of partitions of n*(n+1)/2 into parts not greater than n.
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11
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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
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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FORMULA
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a(n) ~ c * d^n / n^2, where d = 5.4008719041181541524660911191042700520294... = A258234, c = 0.6326058791290010900659134913629203727... . - Vaclav Kotesovec, Sep 07 2014
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MATHEMATICA
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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 *)
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PROG
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(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 ) );
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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