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A341404 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_9)^2 <= n.

%I A341404 #6 Feb 10 2021 20:44:41
%S A341404 1,10,46,130,265,463,799,1339,2014,2780,3860,5444,7301,9263,11783,
%T A341404 15263,19250,23237,27893,34193,41519,48701,56765,67421,79484,91067,
%U A341404 103739,119855,138035,155819,174923,198863,225890,251444,277976,311492,349122,384420,421284
%N A341404 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_9)^2 <= n.
%C A341404 Partial sums of A045851.
%H A341404 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A341404 G.f.: (1 + theta_3(x))^9 / (512 * (1 - x)).
%F A341404 a(n^2) = A055408(n).
%p A341404 b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
%p A341404       b(n, k-1)+add(b(n-j^2, k-1), j=1..isqrt(n))))
%p A341404     end:
%p A341404 a:= proc(n) option remember; b(n, 9)+`if`(n>0, a(n-1), 0) end:
%p A341404 seq(a(n), n=0..38);  # _Alois P. Heinz_, Feb 10 2021
%t A341404 nmax = 38; CoefficientList[Series[(1 + EllipticTheta[3, 0, x])^9/(512 (1 - x)), {x, 0, nmax}], x]
%Y A341404 Cf. A000122, A000606, A003059, A045851, A055408, A055415, A224212, A224213, A302862, A341400, A341401, A341402, A341403, A341405.
%K A341404 nonn
%O A341404 0,2
%A A341404 _Ilya Gutkovskiy_, Feb 10 2021