Morally you should definitely contribute to your DCs childcare costs.
What you need to think about is how much it would cost if YOU had DC 39 weeks of the year, dropped him off at 8.45am and picked him up at 3.15 every day and had to find a job that works around those hours (presumably taking a drop in income to allow for that) or if you can't change jobs you have to pay all the childcare costs for before and after school.
Plus also take on all the homework, packed lunch making, petrol for school runs, costumes for World book day and such like.
At the moment, you are just going to get all the fun parts of parenting; summer holidays, Christmas, Easter, Halloween etc and very little of the hard bits.
For a single parent to work a regular 9-5 job and commute, your son will probably need at least half hour in breakfast club and 2 to 3 hours in after school club. That's around 15 hours of childcare per week for 39 weeks of the year that your ex will be paying.
Assuming a rate of between £5 and £8 an hour (not sure of the average cost in your area) that's between:
15 x 39 x £5 = £2925
And
15 x 39 x £8 = £4680
In childcare costs JUST during term time.
If you're splitting the 13 weeks of school holidays evenly - 6.5 weeks each. I'm presuming you'll take about 4 weeks with your son and pay for 2 weeks childcare. At £50 per day this would be £500 total.
Your ex will probably have to take days off during term time if your son is sick/ teacher training day/ falls over and hits head and needs picking up from school etc. so it's likely she'll max be able to have 3 weeks in holiday time with son. So will then have to pay for 3.5 weeks of holiday childcare- about £1000.
In total that's between £4k and £6k your ex is spending on childcare per annum, Vs your £500.
So yes, you should definitely split costs with her.