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%I #25 Mar 01 2021 06:10:10
%S 1,12,66,220,507,924,1584,2772,4521,6436,8712,12552,18041,23364,28776,
%T 37896,50997,62832,72996,89892,115776,139348,156816,185064,231759,
%U 274044,300828,343564,418638,487080,528528,592284,707421,814836,874170,959508,1128338
%N Number of nonnegative solutions of x1^2 + x2^2 + ... + x12^2 = n.
%H Seiichi Manyama, <a href="/A045853/b045853.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..2000 from T. D. Noe)
%F Coefficient of q^n in (1 + q + q^4 + q^9 + q^16 + q^25 + q^36 + q^49 + q^64 + ...)^12.
%F G.f.: ((1 + theta_3(x)) / 2)^12. - _Ilya Gutkovskiy_, Feb 10 2021
%p b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
%p b(n, k-1)+add(b(n-j^2, k-1), j=1..isqrt(n))))
%p end:
%p a:= b(n, 12):
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Feb 10 2021
%t CoefficientList[((1 + EllipticTheta[3, 0, q])/2)^12 + O[q]^40, q] (* _Jean-François Alcover_, Mar 01 2021 *)
%o (Ruby)
%o def mul(f_ary, b_ary, m)
%o s1, s2 = f_ary.size, b_ary.size
%o ary = Array.new(s1 + s2 - 1, 0)
%o (0..s1 - 1).each{|i|
%o (0..s2 - 1).each{|j|
%o ary[i + j] += f_ary[i] * b_ary[j]
%o }
%o }
%o ary[0..m]
%o end
%o def power(ary, n, m)
%o if n == 0
%o a = Array.new(m + 1, 0)
%o a[0] = 1
%o return a
%o end
%o k = power(ary, n >> 1, m)
%o k = mul(k, k, m)
%o return k if n & 1 == 0
%o return mul(k, ary, m)
%o end
%o def A(k, n)
%o ary = Array.new(n + 1, 0)
%o (0..Math.sqrt(n).to_i).each{|i| ary[i * i] = 1}
%o power(ary, k, n)
%o end
%o p A(12, 100) # _Seiichi Manyama_, May 28 2017
%Y Cf. A010052, A045847.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
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