A317665 Expansion of 1/Sum_{k>=0} x^(k^2).

1, -1, 1, -1, 0, 1, -2, 3, -3, 1, 2, -6, 10, -11, 8, 0, -14, 29, -39, 38, -18, -22, 74, -123, 144, -110, 6, 161, -352, 491, -484, 251, 235, -896, 1528, -1825, 1452, -191, -1892, 4317, -6164, 6243, -3488, -2482, 10788, -18957, 23140, -19085, 3858, 22025, -52833, 77224, -80198, 47899

Offset

0,7

Comments

Convolution inverse of A010052.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Formula

G.f.: 2/(1 + theta_3(q)), where theta_3() is the Jacobi theta function.

a(n) = Sum_{k=0..n} (-1)^k * A337165(n,k).

a(0) = 1; a(n) = -Sum_{k=1..n} A010052(k) * a(n-k). - Seiichi Manyama, Mar 19 2022

Example

G.f. = 1 - x + x^2 - x^3 + x^5 - 2*x^6 + 3*x^7 - 3*x^8 + x^9 + 2*x^10 - 6*x^11 + ...

Maple

a:=series(1/add(x^(k^2), k=0..100), x=0, 54): seq(coeff(a, x, n), n=0..53); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 53; CoefficientList[Series[1/Sum[x^k^2, {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 53; CoefficientList[Series[2/(1 + QPochhammer[x^2]^5/(QPochhammer[x] QPochhammer[x^4])^2), {x, 0, nmax}], x]
nmax = 53; CoefficientList[Series[2/(1 + EllipticTheta[3, 0, q]), {q, 0, nmax}], q]
a[0] = 1; a[n_] := a[n] = -Sum[Boole[IntegerQ[Sqrt[k]]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 53}]

Prog

(PARI) seq(n)=Vec(1/(sum(k=0, sqrtint(n), x^(k^2)) + O(x*x^n))) \\ Andrew Howroyd, Aug 08 2018
(PARI) a(n) = if(n==0, 1, -sum(k=1, n, issquare(k)*a(n-k))); \\ Seiichi Manyama, Mar 19 2022

Crossrefs

Cf. A004402, A006456, A010052, A106507, A323633, A337165.

Sequence in context: A181608 A272873 A295634 * A329315 A109199 A279813

Adjacent sequences: A317662 A317663 A317664 * A317666 A317667 A317668

Keyword

sign

Author

Ilya Gutkovskiy, Aug 08 2018

Status

approved