I am trying to set up a route and have the following routes (core 4510, edge WS-C3750V2-24PS-S  ) (pertinent data in bold):

On the 4510 a show ip route:

     172.18.0.0/24 is subnetted, 2 subnets

C       172.18.2.0 is directly connected, Vlan182

C       172.18.1.0 is directly connected, Vlan15

     172.20.0.0/16 is variably subnetted, 3 subnets, 2 masks

C       172.20.4.4/30 is directly connected, GigabitEthernet9/2

C       172.20.3.0/24 is directly connected, Vlan10

C       172.20.4.0/30 is directly connected, GigabitEthernet9/1

     172.30.0.0/24 is subnetted, 1 subnets

S       172.30.3.0 [1/0] via 172.20.4.2

S*   0.0.0.0/0 [1/0] via 192.168.100.9

C    192.168.0.0/16 is directly connected, Vlan1

On the 3750:

     172.16.0.0/24 is subnetted, 1 subnets

C       172.16.66.0 is directly connected, Vlan12

     172.20.0.0/16 is variably subnetted, 3 subnets, 2 masks

C       172.20.4.4/30 is directly connected, GigabitEthernet2/0/28

C       172.20.3.0/24 is directly connected, Vlan10

C       172.20.4.0/30 is directly connected, GigabitEthernet1/0/28

The two Gigabit ports are fiber attched to the 4510.

I can ping the 172.20.4.2 successfully but I cannot ping 172.20.3.1.  Can someone point out where I may be missing or mis-configured a line?

There is no route for 192.168.100.x network on the 3750 so the 3750 will be unable to respond to the ping. You need to the WS-C3750V2-48PS-S  to get a route for 192.168.100.x.