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

0,3

Links

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

Formula

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

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

Example

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.

Maple

b:= proc(n, i, t) option remember;
      `if`(i<t or n<t*(t+1)/2 or 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))))
    end:
a:= 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[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 *)

Crossrefs

Cf. A161407. - Reinhard Zumkeller, Jun 10 2009

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

Sequence in context: A120892 A195499 A225688 * A184255 A001120 A302629

Adjacent sequences: A109652 A109653 A109654 * A109656 A109657 A109658

Keyword

nonn

Author

Henry Bottomley, Aug 05 2005

Extensions

More terms from Alois P. Heinz, Jan 18 2012

Status

approved