# Square weighted integer partition

As a part of my research I have arrived at the following generating function

$\prod_{k=1}^{\infty}\left(\frac{1}{1-k^{2}x^{k}}\right)$

This is almost similar to the generating function of the integer partition except with the weight $k^2$. One get this generating function by assigning weight equal to product of squares of the partition to each of the partitions. For example n=3 has three partitions 1+1+1, 1+2, 3. We assign weight to each of the partitions $1^2.1^2.1^2=1, 1^2.2^2=4, 3^2=9$. Then add them $1+4+9=14$. This is the coefficient of the 3rd term in the generating function.

I am interested in understanding the asymptotic terms of the generating function. Is this generating function already studied in the literature? If so, can you suggest some place to look at? Or is it possible to relate it to integer partition?

Or any other suggestion will also be helpful.

I have found OEIS page https://oeis.org/A077335 of the above generating function. But there is no reference to asymptotic terms.

Thank you

• I suppose the natural appoach would be to apply saddle-point methods as in chapter VIII of Flajolet & Sedgewick, "Analytic Combinatorics" – Marcel Mar 20 '18 at 13:34
• In A077335 is the formula $\;a(n) \sim c \;3^{2n/3}$ where $c$ depends on $\!\!\mod(n,3)$. Isn't that good enough? What do you mean "asymptotic terms of the g.f."? – Somos Mar 21 '18 at 1:35