A281541 Expansion of Sum_{i>=1} x^(i^2)/(1 - x^(i^2)) / Product_{j>=1} (1 - x^(j^2)).

1, 2, 3, 5, 7, 9, 11, 15, 19, 23, 27, 34, 41, 47, 53, 64, 75, 86, 96, 113, 129, 145, 159, 182, 206, 229, 252, 284, 318, 349, 380, 423, 468, 513, 555, 616, 676, 736, 791, 869, 949, 1026, 1103, 1202, 1310, 1408, 1506, 1631, 1766, 1896, 2020, 2185, 2354, 2525, 2680, 2882, 3094, 3305, 3506, 3751, 4023, 4281, 4537

Offset: 1

Author

Ilya Gutkovskiy, Jan 23 2017

Keywords

• nonn

Comments

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

Convolution of A001156 and A046951.

Examples

			a(8) = 15 because we have [4, 4], [4, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1] and 2 + 5 + 8 = 15.
		

Links

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

Crossrefs

Cf. A000290, A001156, A006128, A046951, A243148.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n], (s->
      `if`(s>n, 0, (p->p+[0, p[1]])(b(n-s, i))))(i^2)+b(n, i-1))
    end:
a:= n-> b(n, isqrt(n))[2]:
seq(a(n), n=1..70);  # Alois P. Heinz, Sep 19 2018

Mathematica

nmax = 63; 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)).

a(n) = Sum_{k=1..n} k * A243148(n,k). - Alois P. Heinz, Sep 19 2018

a(n) ~ exp(3 * 2^(-4/3) * zeta(3/2)^(2/3) * (Pi*n)^(1/3)) * sqrt(Pi/3) / (12*sqrt(n)). - Vaclav Kotesovec, Sep 15 2021