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

0,6

Links

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

Formula

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

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 *)

Prog

(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

Crossrefs

Cf. A037444, A033461, A000009, A000290.

Sequence in context: A275078 A286478 A340756 * A029197 A029174 A058753

Adjacent sequences: A030270 A030271 A030272 * A030274 A030275 A030276

Keyword

nonn

Author

Warren D. Smith

Extensions

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

Status

approved