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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. 5
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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 A117722

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

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Last modified September 20 15:44 EDT 2017. Contains 292275 sequences.