A089333 Number of partitions into a square number of parts.

1, 1, 1, 1, 2, 2, 3, 4, 6, 8, 11, 14, 19, 24, 31, 39, 51, 63, 80, 99, 124, 153, 190, 233, 288, 353, 432, 527, 643, 780, 947, 1145, 1383, 1665, 2002, 2399, 2874, 3431, 4090, 4865, 5779, 6847, 8103, 9568, 11283, 13280, 15610, 18313, 21462, 25108, 29337, 34227

Offset: 0

Author

Vladeta Jovovic, Dec 25 2003

Keywords

• easy
• nonn

Comments

Also number of partitions of n such that the largest part is a square. Example: a(7)=4 because we have [4,3], [4,2,1], [4,1,1,1] and [1,1,1,1,1,1,1]. - Emeric Deutsch, Apr 04 2006

Examples

			a(7)=4 because we have [7], [4,1,1,1], [3,2,1,1] and [2,2,2,1].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Programs

Maple

g:=sum(x^(k^2)/product(1-x^i,i=1..k^2),k=1..7): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=1..51); # Emeric Deutsch, Apr 04 2006
# second Maple program:
b:= proc(n, i) option remember; `if`(n<0, 0,
      `if`(n=0 or i=1, 1, `if`(i<1, 0, b(n, i-1)+
      `if`(i>n, 0, b(n-i, i)))))
    end:
a:= n-> add(b(n-i^2, i^2), i=0..isqrt(n)):
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 24 2015

Mathematica

b[n_, i_] := b[n, i] = If[n < 0, 0, If[n == 0 || i == 1, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i]]]]]; a[n_] := Sum[b[n - i^2, i^2], {i, 0, Sqrt[n]}]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Jan 10 2016, after Alois P. Heinz*)

Formula

G.f.: Sum(x^(n^2)/Product(1-x^i, i = 1 .. n^2), n = 1 .. infinity).

Extensions

a(0)=1 from Alois P. Heinz, Sep 24 2015