A025434 Number of partitions of n into 10 nonzero squares.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 4, 1, 2, 4, 1, 2, 5, 2, 4, 4, 2, 5, 4, 2, 5, 6, 4, 5, 6, 4, 5, 6, 5, 7, 9, 5, 7, 10, 5, 7, 11, 6, 11, 10, 6, 12, 10, 7, 13, 14, 10, 12, 14, 11, 12, 14, 12, 16, 19, 12, 16, 19, 12, 16, 22, 15, 21, 21, 15

Offset: 0

Author

David W. Wilson

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Index entries for sequences related to sums of squares

Crossrefs

Column k=10 of A243148.

Cf. A010052.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
      `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(i^2>n, 0, b(n-i^2, i, t-1))))
    end:
a:= n-> b(n, isqrt(n), 10):
seq(a(n), n=0..120);  # Alois P. Heinz, May 30 2014

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] + If[i^2 > n, 0, b[n - i^2, i, t - 1]]]];
a[n_] := b[n, Sqrt[n] // Floor, 10];
a /@ Range[0, 120] (* Jean-François Alcover, Nov 07 2020, after Alois P. Heinz *)

Formula

a(n) = [x^n y^10] Product_{k>=1} 1/(1 - y*x^(k^2)). - Ilya Gutkovskiy, Apr 19 2019

a(n) = Sum_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} A010052(i) * A010052(j) * A010052(k) * A010052(l) * A010052(m) * A010052(o) * A010052(p) * A010052(q) *A010052(r) * A010052(n-i-j-k-l-m-o-p-q-r). - Wesley Ivan Hurt, Apr 19 2019