A045852 Number of nonnegative solutions of x1^2 + x2^2 + ... + x10^2 = n.

1, 10, 45, 120, 220, 342, 570, 960, 1350, 1640, 2191, 3240, 4200, 4720, 5760, 7920, 9865, 10620, 11965, 15600, 19332, 20550, 22200, 28080, 34200, 35582, 37395, 45720, 54600, 56970, 59460, 71040, 84330, 87090, 88195, 104040, 123200, 125710, 126540, 148560

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2000 from T. D. Noe)

Crossrefs

Cf. A010052, A045847.

Programs

Maple

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+add(b(n-j^2, k-1), j=1..isqrt(n))))
    end:
a:= b(n, 10):
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 10 2021

Mathematica

Take[CoefficientList[Expand[(Total[x^Range[0,5]^2])^10],x],50] (* Harvey P. Dale, May 20 2011 *)

Ruby

def mul(f_ary, b_ary, m)
  s1, s2 = f_ary.size, b_ary.size
  ary = Array.new(s1 + s2 - 1, 0)
  (0..s1 - 1).each{|i|
    (0..s2 - 1).each{|j|
      ary[i + j] += f_ary[i] * b_ary[j]
    }
  }
  ary[0..m]
end
def power(ary, n, m)
  if n == 0
    a = Array.new(m + 1, 0)
    a[0] = 1
    return a
  end
  k = power(ary, n >> 1, m)
  k = mul(k, k, m)
  return k if n & 1 == 0
  return mul(k, ary, m)
end
def A(k, n)
  ary = Array.new(n + 1, 0)
  (0..Math.sqrt(n).to_i).each{|i| ary[i * i] = 1}
  power(ary, k, n)
end
p A(10, 100) # Seiichi Manyama, May 28 2017

Formula

Coefficient of q^n in (1 + q + q^4 + q^9 + q^16 + q^25 + q^36 + q^49 + q^64 + ...)^10.

G.f.: ((1 + theta_3(x)) / 2)^10. - Ilya Gutkovskiy, Feb 10 2021