lacunary sum of binomial coefficients

I am sure there must be a known estimate for sums of the form: $$ S_d(n)=\sum_{j=0}^n \binom{dn}{dj} $$ where $d\ge 1$.

For $d=1$, the answer is obvious.

For $d=2$, the answer is here: Sum with binomial coefficients: $\sum_{k=0}^{n}{2n\choose 2k}$

UPDATE: I am interested in asymptotics. $$ d=3:\quad S_3(n)=\frac 13 8^{n}+(-1)^n \frac 23, $$

$$ d=4:\quad S_4(n)=\frac 14 16^{n}+(-1)^n 2^{2n-1} $$

$$ d=5:\quad S_5(n)=\frac 15 32^n+\frac 25\Biggl(-\frac {11}2+\frac{5\sqrt 5}{2}\Biggr)^n+ \frac 25\Biggl(-\frac {11}2-\frac{5\sqrt 5}{2}\Biggr)^n $$

The leading term is $\frac 1d 2^{dn}$, any bound for the remainder (with a reference:)?

Solution 1:

Let $\omega$ be a primitive $d$ root of unity. Then define

$$\rm a:=\frac{1}{d}\sum_{j=0}^{d-1} \omega^{\,jk}=\begin{cases}1 & \rm k\equiv 0 ~\bmod d \\ 0 & \rm otherwise \end{cases}=\delta_k$$

To see the equality, first check the $\rm k\equiv0$ case, then the $\rm k\equiv 1$ case by symmetry ($\rm a=\omega \cdot a$), then notice the map $\rm j\mapsto \omega^{jk}$ sends the residues modulo $\rm d$ to the $\rm d/gcd(d,k)$th roots of unity, and it is specifically a $\rm \gcd(d,k)$-to-$1$ map, for any nonzero $\rm k$ (hence $\rm a=0$, going mod $\rm d/(d,k)$). (Also see the Kronecker delta function for a definition of the RHS.)

Using the binomial theorem again with interchange of summation, we have

$$\rm \begin{array}{c c}\frac{1}{d}\sum_{j=0}^{d-1} \big(1+\omega^j\big)^{nd} & \rm =\frac{1}{d}\sum_{j=0}^{d-1}\sum_{k=0}^{nd}\binom{nd}{k} \omega^{\,jk} \\ & \rm =\sum_{k=0}^{nd}\binom{nd}{k}\frac{1}{d}\sum_{j=0}^{d-1}\omega^{\,jk} \\ & \rm =\sum_{k=0}^{nd}\binom{nd}{k}\delta_k \\ & \rm =\sum_{\ell=0}^n \binom{nd}{n\ell}. \end{array}$$

Observe that the first leading term will be with $\omega=1$. Set $\rm \omega=e^{2\pi i/d}$; ordering $|1+\omega^j|$ by magnitude, we find the $(v+1)$th leading term is (for $v\le d/2$)

$$\begin{array}{c c} \rm \frac{(1+\omega^v)^{nd}+(1+\omega^{-v})^{nd}}{d} & = \rm \frac{(\omega^{v/2}+\omega^{-v/2})^{nd}(\omega^{vnd/2}+\omega^{-vnd/2})}{d} \\ & =\rm \frac{2}{d}\big(2\cos(2\pi v/d) \big)^{nd}\cos(\pi vnd)\end{array}$$

Similar considerations lead to an offset modular generalization:

$$\rm \frac{1}{d}\sum_{j=0}^{d-1}\omega^{-j\,r} \big(1+\omega^j\big)^{nd} =\sum_{\ell=0}^n \binom{nd}{n\ell+r}.$$

Solution 2:

It is well known that

$$\sum_{j=0}^{n} \binom{n}{dj} = \frac{1}{d} \sum_{j=0}^{d-1} (1 + w^j)^n$$

where $w$ is a primitive $d^{th}$ root of unity.

This you can get by considering

$$ (1+z)^n = \sum_{j=0}^{n} \binom{n}{j} z^j$$

Set $z = w^j$ for $j= 0$ to $d-1$ and add up.