VLAN not allowed in management domain

If i understand your layout correctly, it sounds like VTP prunning is filtering this traffic between switches since the AS-01 switch does not have any ports that are members of vlan370. Do you have VTP and VTP pruning enabled on your switches? Also check your trunks on your switches to see if 370 is allowed on each.
If you are not using VTP then make sure vlan 370 is configured on all switches and allowed across the proper trunks.

It looks like there are some VTP inconsistencies across the network. Below is the VTP info for the AS-01 switch (the uplink switch) and the AS-02 switch the actual access switch for this server.  I put X's over the VTP domain mode for security reasons.  Once again VLAN 370 shows up as allowed on the trunks just not "allowed and active in management domain" as shown above.  What is weired though is that ALL 3 switches CAN ping the default gateway.  Just not the server.

Here is the VTP info on AS-01

VTP Version                     : running VTP1 (VTP2 capable)
Configuration Revision          : 0
Maximum VLANs supported locally : 1005
Number of existing VLANs        : 14
VTP Operating Mode              : Transparent
VTP Domain Name                 : xxxx
VTP Pruning Mode                : Enabled
VTP V2 Mode                     : Disabled
VTP Traps Generation            : Disabled
MD5 digest                      : 0xD7 0x0F 0x5B 0x48 0x18 0x43 0x97 0xC8
Configuration last modified by 0.0.0.0 at 0-0-00 00:00:00


Here is the VTP info on AS-02 the access switch for the server:

VTP Version                     : running VTP1 (VTP2 capable)
Configuration Revision          : 23
Maximum VLANs supported locally : 1005
Number of existing VLANs        : 15
VTP Operating Mode              : Client
VTP Domain Name                 : xxxx
VTP Pruning Mode                : Enabled
VTP V2 Mode                     : Disabled
VTP Traps Generation            : Disabled
MD5 digest                      : 0xE1 0x7D 0xB8 0x30 0xB7 0x7A 0x25 0xFA
Configuration last modified by 10.200.30.1 at 2-15-10 15:48:10

A little more troubleshooting.  I went ahead and made a switch port on the AS-01 access the VLAN 370.  Of course this then adds it to the vlan info (database) for this switch, thereby allowing the PINGS to cross this switches trunk interfaces from AS-02 to the core switch.  
How do I add a VLAN on a lower switch that needs to get OUT without having to add an access port in that new VLAN on the intermediate switch????
And of course as seen in the above VTP status configs, the AS-01 switch has 14 existing VLANs while the AS-02 switch has 15 VLANS.  Of course, they both now have 15.  So I can now ping across, I just didn't want to add this VLAN to the AS-01 switch as an access port.
Does this make sense?  

Yes, the core is in server mode.  I definitely don't want an AS switch to be in server mode.  So client is probably best.  
Thanks,

It dosent hurt to have 2 switches in server mode. That way if one goes down you still have one managing VTP. Just keep in mind that whichever server has the latest version will update all others with its vlan database.
My suggestion would be change it to client first. Let it update its vlan db then change it to server. But with 3 switches its not a big deal.

Makes sense.  We have 3 core switches in the network - 1 per building.  This happens to be a core switch at one of our school buildings.