A281704 Expansion of Sum_{i>=1} x^(i^2) / (1 - Sum_{j>=1} x^(j^2))^2.

1, 2, 3, 5, 9, 15, 23, 35, 55, 87, 134, 202, 305, 463, 700, 1049, 1565, 2334, 3478, 5168, 7654, 11314, 16705, 24632, 36260, 53295, 78237, 114728, 168059, 245916, 359483, 525021, 766144, 1117107, 1627587, 2369609, 3447549, 5012588, 7283577, 10577198, 15351519, 22268890, 32286666, 46788056, 67770831

Offset: 1

Author

Ilya Gutkovskiy, Jan 30 2017

Keywords

• nonn

Comments

Total number of parts in all compositions (ordered partitions) of n into squares (A000290).

Examples

			a(6) = 15 because we have [4, 1, 1], [1, 4, 1], [1, 1, 4], [1, 1, 1, 1, 1, 1] and 3 + 3 + 3 + 6 = 15.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..6575
• Index entries for sequences related to sums of squares
• Index entries for sequences related to compositions

Crossrefs

Cf. A000290, A006456, A337165.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, [1, 0], add(
      (p-> p+[0, p[1]])(b(n-j^2)), j=1..isqrt(n)))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=1..45);  # Alois P. Heinz, Aug 07 2019

Mathematica

nmax = 45; Rest[CoefficientList[Series[Sum[x^i^2, {i, 1, nmax}]/(1 - Sum[x^j^2, {j, 1, nmax}])^2, {x, 0, nmax}], x]]
nmax = 45; Rest[CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)/(2 (1 + (1 - EllipticTheta[3, 0, x])/2)^2), {x, 0, nmax}], x]]

Formula

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

a(n) = Sum_{k=0..n} k * A337165(n,k). - Alois P. Heinz, Feb 03 2021