A341405 - OEIS

%I #5 Feb 10 2021 20:45:38

%S 1,11,56,176,396,738,1308,2268,3618,5258,7449,10689,14889,19609,25369,

%T 33289,43154,53774,65739,81339,100671,121221,143421,171501,205701,

%U 241283,278678,324398,378998,435968,495428,566468,650798,737888,826083,930123,1053323

%N Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_10)^2 <= n.

%C Partial sums of A045852.

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>

%F G.f.: (1 + theta_3(x))^10 / (1024 * (1 - x)).

%F a(n^2) = A055409(n).

%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:= proc(n) option remember; b(n, 10)+`if`(n>0, a(n-1), 0) end:

%p seq(a(n), n=0..36);  # _Alois P. Heinz_, Feb 10 2021

%t nmax = 36; CoefficientList[Series[(1 + EllipticTheta[3, 0, x])^10/(1024 (1 - x)), {x, 0, nmax}], x]

%Y Cf. A000122, A000606, A003059, A045852, A055409, A055416, A224212, A224213, A302862, A341400, A341401, A341402, A341403, A341404.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Feb 10 2021