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A303909 Expansion of 2*(1 - x)/(3 - theta_3(x)), where theta_3() is the Jacobi theta function. 0

%I

%S 1,0,0,0,1,1,1,1,2,4,5,6,8,13,19,26,36,51,74,105,148,208,296,421,597,

%T 846,1198,1699,2409,3417,4843,6865,9732,13799,19566,27739,39325,55749,

%U 79041,112063,158877,225241,319331,452734,641866,910001,1290137,1829079,2593169,3676457,5212266

%N Expansion of 2*(1 - x)/(3 - theta_3(x)), where theta_3() is the Jacobi theta function.

%C First differences of A006456.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>

%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>

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

%F G.f.: (1 - x)/(1 - Sum_{k>=1} x^(k^2)).

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

%p `if`(n=0, 1, add(b(n-j^2), j=1..isqrt(n))))

%p end:

%p a:= n-> b(n)-`if`(n=0, 0, b(n-1)):

%p seq(a(n), n=0..60); # _Alois P. Heinz_, May 02 2018

%t nmax = 50; CoefficientList[Series[2 (1 - x)/(3 - EllipticTheta[3, 0, x]), {x, 0, nmax}], x]

%t nmax = 50; CoefficientList[Series[(1 - x)/(1 - Sum[x^k^2, {k, 1, nmax}]), {x, 0, nmax}], x]

%t a[0] = 1; a[n_] := a[n] = Sum[Boole[IntegerQ[k^(1/2)]] a[n - k], {k, 1, n}]; Differences[Table[a[n], {n, -1, 50}]]

%Y Cf. A000290, A006456, A078134, A303667.

%K nonn

%O 0,9

%A _Ilya Gutkovskiy_, May 02 2018

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Last modified June 6 18:59 EDT 2020. Contains 334832 sequences. (Running on oeis4.)