A109655 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.

1, 1, 3, 8, 33, 141, 676, 3370, 17575, 94257, 517971, 2900900, 16509188, 95220378, 555546058, 3273480400, 19456066175, 116521302221, 702567455381, 4261765991164, 25992285913221, 159303547578873, 980701254662294, 6061894625462492, 37609015174472628

Offset: 0

Author

Henry Bottomley, Aug 05 2005

Keywords

• nonn

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A161407. - Reinhard Zumkeller, Jun 10 2009

Row n=3 of A204459. - Alois P. Heinz, Jan 18 2012

Programs

Maple

b:= proc(n, i, t) option remember;
      `if`(it*(2*i-t+1)/2, 0,
      `if`(n=0, 1, b(n, i-1, t) +`if`(n b(n*(3*n+1)/2, 3*n, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 18 2012

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[it*(2*i-t+1)/2, 0, If[n == 0, 1, b[n, i-1, t] + If[nJean-François Alcover, Oct 05 2015, after Alois P. Heinz *)

Formula

a(n) = A067059(n,2n) = A067059(2n,n).

Slightly less than but close to (27/4)^n*sqrt(3)/(2*Pi*n^2).

Extensions

More terms from Alois P. Heinz, Jan 18 2012