A342228 Total sum of parts which are squares in all partitions of n.

0, 1, 2, 4, 11, 16, 27, 42, 69, 108, 158, 229, 334, 469, 656, 903, 1255, 1685, 2283, 3032, 4033, 5290, 6936, 8986, 11650, 14969, 19172, 24402, 30998, 39110, 49260, 61712, 77155, 96000, 119209, 147394, 181958, 223713, 274533, 335792, 409980, 498981, 606273, 734572

Offset

0,3

Links

Table of n, a(n) for n=0..43.

Formula

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

a(n) = Sum_{k=1..n} A035316(k) * A000041(n-k).

Example

For n = 4 we have:
---------------------------------
Partitions        Sum of parts
.              which are squares
---------------------------------
4 ................... 4
3 + 1 ............... 1
2 + 2 ............... 0
2 + 1 + 1 ........... 2
1 + 1 + 1 + 1 ....... 4
---------------------------------
Total .............. 11
So a(4) = 11.

Mathematica

nmax = 43; CoefficientList[Series[Sum[k^2 x^(k^2)/(1 - x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}]/Product[(1 - x^j), {j, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[DivisorSum[k, # &, IntegerQ[#^(1/2)] &] PartitionsP[n - k], {k, 1, n}], {n, 0, 43}]

Crossrefs

Cf. A000041, A000290, A035316, A066186, A073336, A342229.

Sequence in context: A180384 A023168 A134419 * A024819 A194545 A329800

Adjacent sequences: A342225 A342226 A342227 * A342229 A342230 A342231

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 06 2021

Status

approved