Here is a fairly simplistic piece of code that follows the approach given in the Guide to DHS Statistics, and produces estimates for five five-year periods.
It doesn't produce standard errors or confidence intervals, but allows you to see how the calculations are done:

* Example of early childhood mortality rates calculations
* Trevor Croft, March 9, 2018

* Change directory to the data directory
cd "C:\Users\xxxx\Data"
* Open DHS dataset - births recode file
use v005 v008 b3 b5 b7 using "IABR71FL.DTA", clear

* Create variables for time period limits - need to use variables as these change from case to case
gen t1 = .
gen t2 = .
* Initialize local variable lists used later
local vlist
local vlist2

* Loop through 5-year time periods
forvalues period = 0/4 {

	* Calculate upper limit of time period
	replace t2 = v008 - 60*`period'
	* Calculate lower limit of time period
	replace t1 = t2 - 60
	* List age group lower limits
	local agegroups 0 1 3 6 12 24 36 48 60
	* Turn thse into tokens to use for the upper limits of the age groups
	tokenize `agegroups'
	* Loop through the age groups
	foreach age of numlist `agegroups' {
		* Ignore the 60+ age group - this was just to set the upper limit for the last age group - see a2
		if (`age' < 60) {
			* Create local for lower limit of age group - use locals as these are constants
			local a1 = `age'
			* Create local for upper limit of age group = the lower limit of the next age group
			local a2 = `2'
			
			* Cohort A numerator
			gen numA`age'_`period' = ((`a1' <= b7 & b7 < `a2') & (t1 - `a2' <= b3 & b3 < t1 - `a1'))
			* Cohort B numerator
			gen numB`age'_`period' = ((`a1' <= b7 & b7 < `a2') & (t1 - `a1' <= b3 & b3 < t2 - `a2'))
			* Cohort C numerator
			gen numC`age'_`period' = ((`a1' <= b7 & b7 < `a2') & (t2 - `a2' <= b3 & b3 < t2 - `a1'))
			* Cohort A denominator
			gen denA`age'_`period' = ( (b5 == 1 | `a1' <= b7)  & (t1 - `a2' <= b3 & b3 < t1 - `a1'))
			* Cohort B denominator
			gen denB`age'_`period' = ( (b5 == 1 | `a1' <= b7)  & (t1 - `a1' <= b3 & b3 < t2 - `a2'))
			* Cohort C denominator
			gen denC`age'_`period' = ( (b5 == 1 | `a1' <= b7)  & (t2 - `a2' <= b3 & b3 < t2 - `a1'))
			
			* Count half for deaths for cohort C, except for the last period where all deaths are counted
			local f = 0.5
			if (`period' == 0) {
				local f = 1
			}
			* Sum numerators from cohorts A, B and C for this case
			gen num`age'_`period' = 0.5*numA`age'_`period' + numB`age'_`period' + numC`age'_`period'*`f'
			* Sum denominators from chorts A, B and C for this case
			gen den`age'_`period' = 0.5*denA`age'_`period' + denB`age'_`period' + denC`age'_`period'*0.5
			
			* Generate list of numerator and denominator variables for period and age for collapse command below
			local vlist `vlist' num`age'_`period' den`age'_`period'
			* Similarly generate list of numerator and denominator variables for period only for reshape command below
			if (`period' == 0) {
				local vlist2 `vlist2' num`age'_  den`age'_
			}
		}
		* Shift the token list to the next age group 
		mac shift
	}
}


* Sum all numerators and denominators - weighted sum
collapse (sum) `vlist' [pw=v005/1000000]

* Add a variable to act as ID for the reshape
gen x = 0
* Reshape long by age group
reshape long `vlist2', i(x) j(period)
* Drop the underscore (_) on the end of variable names
rename *_ *

* Reshape now for periods
reshape long num den, i(period) j(a1)
* Drop the x variable as we no longer need it
drop x

* Generate the upper bounds of the age groups
gen a2 = a1[_n+1]
replace a2 = 60 if a1 == 48

* Calculate the age group mortality probabilities
gen death = num / den
* Calculate the age group survival probabilities
gen surv = 1 - death

* Generate product of survival probabilities:
gen prodsurv = surv if a1 == 0
replace prodsurv = surv * prodsurv[_n-1] if a1 > 0
* Generate product of survival probabilities for child mortality rate, starting at 12 months
gen prodsurv2 = surv if a1 == 12
replace prodsurv2 = surv * prodsurv2[_n-1] if a1 > 12

* Neonatal mortality rate
gen nmr = 1000*(1-prodsurv) if a2 == 1
* Postneonatal mortality rate (calculated later)
gen pnmr = .
* Infant mortality rate
gen imr = 1000*(1-prodsurv) if a2 == 12
* Child mortality rate
gen cmr = 1000*(1-prodsurv2) if a2 == 60
* Under-five mortality rate
gen u5mr = 1000*(1-prodsurv) if a2 == 60

* Capture just the rates
collapse (min) nmr pnmr imr cmr u5mr, by(period)

* Postneonatal mortality rate = IMR - NMR
replace pnmr = imr - nmr

* Now see the results
list

And the results basically match the syncmrates program

     +---------------------------------------------------------------+
     | period        nmr       pnmr        imr        cmr       u5mr |
     |---------------------------------------------------------------|
  1. |      0   29.46365    11.2654   40.72905   9.390652   49.73727 |
  2. |      1   31.49295   12.24667   43.73962   11.31612   54.56078 |
  3. |      2   33.03296   13.47327   46.50623   12.88736   58.79426 |
  4. |      3   36.41945   15.01405    51.4335   16.21401   66.81353 |
  5. |      4   40.38089   18.37582   58.75671   19.31465   76.93649 |
     +---------------------------------------------------------------+