A030273 Number of partitions of n^2 into distinct squares.

1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 3, 4, 2, 7, 8, 12, 13, 16, 25, 28, 55, 51, 91, 90, 158, 176, 288, 297, 487, 521, 847, 908, 1355, 1580, 2175, 2744, 3636, 4452, 5678, 7385, 9398, 11966, 14508, 19322, 23065, 31301, 36177, 49080, 57348, 77446, 91021, 121113, 141805

Offset: 0

Author

Warren D. Smith

Keywords

• nonn

Links

• Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 (terms 0..750 from Alois P. Heinz)

Crossrefs

Cf. A037444, A033461, A000009, A000290.

Programs

Haskell

a030273 n = p (map (^ 2) [1..]) (n^2) where
   p _  0 = 1
   p (k:ks) m | m < k     = 0
              | otherwise = p ks (m - k) + p ks m
-- Reinhard Zumkeller, Aug 14 2011

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(n>i*(i+1)*(2*i+1)/6, 0, b(n, i-1)+
      `if`(i^2>n, 0, b(n-i^2, i-1))))
    end:
a:= n-> b(n^2, n):
seq(a(n), n=0..50);  # Alois P. Heinz, Nov 20 2012

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, If[n > i*(i+1)*(2*i+1)/6, 0, b[n, i-1] +If[i^2 > n, 0, b[n-i^2, i-1]]]]; a[n_] := b[n^2, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 30 2015, after Alois P. Heinz *)

Formula

a(n) = [x^(n^2)] Product_{k>=1} (1 + x^(k^2)). - Ilya Gutkovskiy, Apr 13 2017

Extensions

a(0)=1 prepended by Alois P. Heinz, Feb 18 2015