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%I #24 Oct 05 2015 10:55:15
%S 1,1,3,8,33,141,676,3370,17575,94257,517971,2900900,16509188,95220378,
%T 555546058,3273480400,19456066175,116521302221,702567455381,
%U 4261765991164,25992285913221,159303547578873,980701254662294,6061894625462492,37609015174472628
%N Number of partitions of n^2 into up to n parts each no more than 2n, or of n(3n+1)/2 into exactly n distinct parts each no more than 3n.
%H Alois P. Heinz, <a href="/A109655/b109655.txt">Table of n, a(n) for n = 0..100</a>
%F a(n) = A067059(n,2n) = A067059(2n,n).
%F Slightly less than but close to (27/4)^n*sqrt(3)/(2*Pi*n^2).
%e a(3) = 8 since 3^2=9 can be partitioned into 3+3+3, 4+3+2, 4+4+1, 5+4, 5+3+1, 5+2+2, 6+3, or 6+2+1, while 3*(3*3+1)/2=15 can be partitioned into 6+5+4, 7+5+3, 7+6+2, 8+6+1, 8+5+2, 8+4+3, 9+5+1, or 9+4+2.
%p b:= proc(n, i, t) option remember;
%p `if`(i<t or n<t*(t+1)/2 or n>t*(2*i-t+1)/2, 0,
%p `if`(n=0, 1, b(n, i-1, t) +`if`(n<i, 0, b(n-i, i-1, t-1))))
%p end:
%p a:= n-> b(n*(3*n+1)/2, 3*n, n):
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Jan 18 2012
%t b[n_, i_, t_] := b[n, i, t] = If[i<t || n<t*(t+1)/2 || n>t*(2*i-t+1)/2, 0, If[n == 0, 1, b[n, i-1, t] + If[n<i, 0, b[n-i, i-1, t-1]]]]; a[n_] := b[n*(3*n+1)/2, 3*n, n]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Oct 05 2015, after _Alois P. Heinz_ *)
%Y Cf. A161407. - _Reinhard Zumkeller_, Jun 10 2009
%Y Row n=3 of A204459. - _Alois P. Heinz_, Jan 18 2012
%K nonn
%O 0,3
%A _Henry Bottomley_, Aug 05 2005
%E More terms from _Alois P. Heinz_, Jan 18 2012
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