A281542 Expansion of Sum_{i>=1} x^(i^2)/(1 + x^(i^2)) * Product_{j>=1} (1 + x^(j^2)).

1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 0, 0, 2, 3, 0, 1, 2, 0, 0, 2, 3, 0, 0, 0, 3, 5, 0, 0, 5, 7, 0, 0, 0, 2, 3, 1, 2, 3, 4, 2, 5, 3, 0, 0, 5, 7, 0, 0, 4, 9, 4, 2, 5, 7, 5, 3, 4, 2, 3, 0, 5, 10, 4, 1, 11, 12, 0, 2, 6, 7, 4, 0, 2, 12, 12, 0, 6, 15, 9, 2, 8, 7, 3, 7, 8, 10, 9, 5, 8, 21, 13, 0, 7, 19, 13, 0, 2, 10, 13, 8

Offset: 1

Author

Ilya Gutkovskiy, Jan 23 2017

Keywords

• nonn

Comments

Total number of parts in all partitions of n into distinct squares.

Examples

			a(26) = 5 because we have [25, 1], [16, 9 ,1] and 2 + 3 = 5.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Index entries for related partition-counting sequences

Crossrefs

Cf. A000290, A001422, A015723, A033461, A317529, A341040.

Programs

Mathematica

nmax = 100; Rest[CoefficientList[Series[Sum[x^i^2/(1 + x^i^2), {i, 1, nmax}] Product[1 + x^j^2, {j, 1, nmax}], {x, 0, nmax}], x]]

Formula

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

From Alois P. Heinz, Feb 03 2021: (Start)

a(n) = Sum_{k>=0} k * A341040(n,k).

a(n) = 0 <=> n in { A001422 }. (End)