eulerian numbers -- A011818 Extension and Variant

[Paul D. Hanna]

Seqfans, 
      Due to the nice dicussion between Mitch and Gottfried, 
I noticed that A011818 needs extending. 
A formula is given using Euler numbers A008292, 
but only a very few terms (5) show in OEIS. 
 
Also, I give a variant of A011818 below 
that should be included in OEIS. 
If a researcher arrived at the sequence by counts or measures, 
the formula may be very difficult to deduce from the numbers alone. 
  
I will submit these in a few weeks. 
   Paul
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A011818  
Normalized volume of center slice of n-dimensional cube: 
2^n * n! * Vol({ x\in [ 0,1 ]: sum_{i=1}^n n/2 <= x_i <= (n+1)/2 }).  
 
1, 3, 16, 115, 1056 
  
FORMULA:  
V(d) = sum_{k=1}^{d-1} {d\choose k-1} A_{d, k} 
where A_{k, d} denotes the Eulerian number 
(permutations of a d-set with k-1 descents). 
 
RESTATED:
A011818(n) = Sum_{k=1..n} C(n,k-1)*A008292(n,k) for n>=1.
  
EXTENSION: 
1,3,16,115,1056,11774,154624,2337507,39984640,763546234,16101629952,
371644257582,9319104528384,252270887452380,7332475985461248,
227761317947788323,7529455986838732800,263948439074152148450,
 
EXAMPLE: 
1 = 1*1
3 = 1*1 + 2*1
16 = 1*1 + 3*4 + 3*1
115 = 1*1 + 4*11 + 6*11 + 4*1
...
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NEW SEQUENCE. 
In light of A011818, it seems plausible that 
the following variant may naturally arise. 
  
NAME: 
Sum_{k=1..n} C(n-1,k-1)*A008292(n,k) for n>=1.
 
1,2,10,68,606,6612,85492,1277096,21641590,410144180,8595133548,
197346180792,4926442358124,132847425483528,3848398710032616,
119187270233781456,3929892162743796390,137444081992905303540,
 
EXAMPLE:
1 = 1*1
2 = 1*1 + 1*1
10 = 1*1 + 2*4 + 1*1
68 = 1*1 + 3*11 + 3*11 + 1*1
...
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END