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A301335 a(n) = [x^n] 1/(1 + (1/2)*n*(1 - theta_3(x))), where theta_3() is the Jacobi theta function. 2

%I

%S 1,1,4,27,260,3175,47304,833147,16941120,390611331,10070060200,

%T 287028156162,8962583345856,304255011200647,11156593415089808,

%U 439452231820920000,18505340390664634384,829599437871129843839,39447684087807950938908,1983038000428208822539998,105080571577382659860160800

%N a(n) = [x^n] 1/(1 + (1/2)*n*(1 - theta_3(x))), where theta_3() is the Jacobi theta function.

%C Number of compositions (ordered partitions) of n into squares of n kinds.

%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 a(n) = [x^n] 1/(1 - n*Sum_{k>=1} x^(k^2)).

%F a(n) ~ n^n * (1 + 1/n^2 - 3/n^3 + 1/(2*n^4) - 13/(2*n^5) + 127/(6*n^6) - 4/n^7 + 335/(8*n^8) - 665/(4*n^9) + 337/(15*n^10) + ...). - _Vaclav Kotesovec_, Mar 19 2018

%t Table[SeriesCoefficient[1/(1 + (1/2) n (1 - EllipticTheta[3, 0, x])), {x, 0, n}], {n, 0, 20}]

%t Table[SeriesCoefficient[1/(1 - n Sum[x^k^2, {k, 1, n}]), {x, 0, n}], {n, 0, 20}]

%Y Cf. A000290, A006456, A240944, A300974, A301334.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Mar 18 2018

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Last modified August 1 15:22 EDT 2021. Contains 346393 sequences. (Running on oeis4.)