E.G.F. Challenge for New Triangle A102316

[Paul D. Hanna]

Here is a nice E.G.F.-challenge to all of you Exceptionally Gifted Folks
... 
 
Valery Liskovets has a nice sequence (along with formula): 
http://www.research.att.com/projects/OEIS?Anum=A082161
1,3,16,127,1363,18628,311250,6173791,142190703,3737431895,
110577492346,3641313700916,132214630355700,5251687490704524,
226664506308709858 
 
I have by chance found a triangle that is related to A082161 
with a recurrence that is begging for a nice E.G.F. to come along ... 
 
The triangle is as follows:
1,
1,1,
1,3,3,
1,7,16,16,
1,15,63,127,127,
1,31,220,728,1363,1363,
1,63,723,3635,10450,18628,18628,
1,127,2296,16836,69086,180854,311250,311250,
1,255,7143,74487,419917,1505041,3683791,6173791,6173791,
1,511,21940,319888,2419473,11449719,37236256,86626584,142190703,142190703
,
...
and has the simple recurrence:  

T(n,k) = T(n,k-1) + (k+1)*T(n-1,k) for n>k>0, 
with T(n,0)=1 for n>=0 and T(n,n)=T(n,n-1) for n>0. 

EXAMPLE.
T(5,2) = 220 = 1*1 + 2*15 + 3*63 = 1*T(4,0) + 2*T(4,1) + 3*T(4,2).
T(5,2) = 220 = 31 + 3*63 = T(5,1) + (2+1)*T(4,2).
T(5,3) = 728 = 220 + 4*127 = T(5,2) + (3+1)*T(4,3).
  
I just now submitted this triangle to the OEIS as A102316.

I believe that this triangle should have an interesting E.G.F. ... 
can anyone find one? 
 
Thanks,
       Paul
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