Estimating $\sum\limits_{d\mid n}{d+a\choose b}$

Question

Is there any way of estimating a sum like $$\sum_{d\mid n}{d+a\choose b},$$ for positive integers $a$ and $b$? For example, in the OEIS we find that

$$\begin{align*} \sum_{d\mid n}{d+1\choose 2} &=\frac{1}{2}(\sigma_1(n)+\sigma_2(n)),\\ \sum_{d\mid n}{d+2\choose 3} &=\frac{1}{6}(2\sigma_1(n)+3\sigma_2(n)+\sigma_3(n)),\\ \sum_{d\mid n}{d+3\choose 4} &=\frac{1}{24}(6\sigma_1(n)+11\sigma_2(n)+6\sigma_3(n)+\sigma_4(n)). \end{align*}$$

I fail to spot the pattern for $d+4$. Anyway, what I'm looking for is more of a simple asymptotic expression (even though the above examples are kinda cute!).

Answer 1

$\sum_{d\mid n}{d+a\choose b} $

The unsigned Stirling numbers of the first kind, $c(n, k)$ are defined by $\prod_{k=0}^{n-1} (x+k) =\sum_{k=0}^n c(n, k)x^k $. (see https://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind)

Therefore

$\begin{array}\\ \rho_{a, b}(n) &=\sum_{d\mid n}{d+a\choose b}\\ &=\sum_{d\mid n}{d+a\choose b}\\ &=\dfrac1{b!}\sum_{d\mid n}\dfrac{(d+a)!}{(d+a-b)!}\\ &=\dfrac1{b!}\sum_{d\mid n}\prod_{k=a-b+1}^a (d+k)\\ &=\dfrac1{b!}\sum_{d\mid n}\prod_{k=0}^{b-1} (d+a-b+1+k)\\ &=\dfrac1{b!}\sum_{d\mid n}\sum_{k=0}^{b}(d+a-b+1)^kc(b, k)\\ &=\dfrac1{b!}\sum_{d\mid n}\sum_{k=0}^{b}c(b, k)\sum_{j=0}^k\binom{k}{j}d^j(a-b+1)^{k-j}\\ &=\dfrac1{b!}\sum_{d\mid n}\sum_{k=0}^{b}c(b, k)\sum_{j=0}^k\binom{k}{j}d^jr^{k-j} \quad r = a-b+1\\ &=\dfrac1{b!}\sum_{k=0}^{b}c(b, k)\sum_{j=0}^k\binom{k}{j}r^{k-j}\sum_{d\mid n}d^j\\ &=\dfrac1{b!}\sum_{k=0}^{b}c(b, k)\sum_{j=0}^k\binom{k}{j}r^{k-j}\sigma_j(n)\\ &=\dfrac1{b!}\sum_{j=0}^n\sigma_j(n)\sum_{k=j}^{b}c(b, k)\binom{k}{j}r^{k-j}\\ &=\dfrac1{b!}\sum_{j=0}^n\sigma_j(n)\sum_{k=0}^{b-j}c(b, k+j)\binom{k+j}{j}r^{k}\\ &=\dfrac1{b!}\sum_{j=0}^n\sigma_j(n)u(a, b, j)\quad\text{where } u(a, b, j)=\sum_{k=0}^{b-j}c(b, k+j)\binom{k+j}{j}r^{k}\\ \end{array} $

If $b=a+1$, as in the examples, $r=0$ so $u(a, a+1, j) =\sum_{k=0}^{b-j}c(a+1, k+j)\binom{k+j}{j}r^{k} =c(a+1, j)\binom{j}{j} =c(a+1, j) $ so that $\rho(a, a+1, n) =\dfrac1{(a+1)!}\sum_{j=0}^n\sigma_j(n)u(a, a+1, j) =\dfrac1{(a+1)!}\sum_{j=0}^n\sigma_j(n)c(a+1, j) $.

Answer 2

This is a complement to @marty cohen answer. We can also give an identity for the case $a=-1$.

With the notation ${n\brack k}=c(n,k)$, we have $$ \sum_{d\mid n}{d-1\choose b} = \frac{1}{b!}\sum_{j=0}^b{b+1\brack j+1}(-1)^{b-j}\sigma_j(n). $$