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

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, 846, 1198, 1699, 2409, 3417, 4843, 6865, 9732, 13799, 19566, 27739, 39325, 55749, 79041, 112063, 158877, 225241, 319331, 452734, 641866, 910001, 1290137, 1829079, 2593169, 3676457, 5212266

Offset: 0

Author

Ilya Gutkovskiy, May 02 2018

Keywords

• nonn

Comments

First differences of A006456.

Links

• Eric Weisstein's World of Mathematics, Jacobi Theta Functions
• Index entries for sequences related to compositions
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A000290, A006456, A078134, A303667.

Programs

Maple

b:= proc(n) option remember; `if`(n<0, 0,
      `if`(n=0, 1, add(b(n-j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n)-`if`(n=0, 0, b(n-1)):
seq(a(n), n=0..60);  # Alois P. Heinz, May 02 2018

Mathematica

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

Formula

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