A276559 Expansion of Sum_{k>=1} k^2*x^k^2/(1 - x^k^2) * Product_{k>=1} 1/(1 - x^k^2).

1, 2, 3, 8, 10, 12, 14, 24, 36, 40, 44, 60, 78, 84, 90, 128, 153, 180, 190, 240, 273, 308, 322, 384, 475, 520, 567, 644, 754, 810, 868, 992, 1122, 1258, 1330, 1548, 1702, 1862, 1950, 2200, 2460, 2646, 2838, 3124, 3510, 3726, 3948, 4320, 4802, 5200, 5457, 6032, 6572, 7128, 7425, 8064, 8778, 9454, 9971, 10680

Offset: 1

Author

Ilya Gutkovskiy, Apr 10 2017

Keywords

• nonn

Comments

Sum of all parts of all partitions of n into squares.

Convolution of the sequences A001156 and A035316.

Examples

			a(8) = 24 because we have [4, 4], [4, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1] and 3*8 = 24.
		

Links

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

Crossrefs

Cf. A000290, A001156, A035316, A066186, A243148, A281541.

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]*s])(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 = 60; Rest[CoefficientList[Series[Sum[k^2 x^k^2/(1 - x^k^2), {k, 1, nmax}] Product[1/(1 - x^k^2), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 60; Rest[CoefficientList[Series[x D[Product[1/(1 - x^k^2), {k, 1, nmax}], x], {x, 0, nmax}], x]]

Formula

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

G.f.: x*f'(x), where f(x) = Product_{k>=1} 1/(1 - x^k^2).

a(n) = n * A001156(n).

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