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Expansion of (theta_3(x) - 1)^2 / (2 * (3 - theta_3(x))).
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%I #12 Sep 14 2021 20:48:20

%S 1,1,1,3,4,5,7,10,16,22,30,43,62,88,123,175,249,354,502,710,1006,1427,

%T 2024,2869,4068,5767,8176,11593,16436,23301,33033,46832,66398,94137,

%U 133461,189211,268252,380315,539192,764433,1083764,1536498,2178364,3088365,4378502,6207581,8800750

%N Expansion of (theta_3(x) - 1)^2 / (2 * (3 - theta_3(x))).

%C Number of compositions (ordered partitions) of n into two or more squares.

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

%F From _Alois P. Heinz_, Sep 14 2021: (Start)

%F a(n) = A006456(n) - A010052(n).

%F a(n) = Sum_{k=2..n} A337165(n,k). (End)

%p b:= proc(n, t) option remember; `if`(n=0, `if`(t=0, 1, 0), add((

%p s->`if`(s>n, 0, b(n-s, max(0, t-1))))(j^2), j=1..isqrt(n)))

%p end:

%p a:= n-> b(n, 2):

%p seq(a(n), n=2..48); # _Alois P. Heinz_, Sep 14 2021

%t nmax = 48; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^2/(2 (3 - EllipticTheta[3, 0, x])), {x, 0, nmax}], x] // Drop[#, 2] &

%Y Cf. A000290, A006456, A010052, A033985, A219610, A337165, A347806, A347807, A347808, A347809.

%K nonn

%O 2,4

%A _Ilya Gutkovskiy_, Sep 14 2021