TITLE: generalizing Wilf's conjecture: Uppuluri-Carpenter numbers
QUESTION [8 upvotes]: The complementary Bell numbers have the exponential generating function
$$\sum_{n\geq0}\tilde{B}_nx^n=e^{1-e^x}.$$
Herb Wilf conjectured that $\tilde{B}_n=0$ only for $n=2$. By now, there are a few papers with a proof.
I wish to generalize this in the same spirit. Let's introduce super exponentials recursively by $e(x)=e^{(0)}(x):=1-e^x$ and $e^{(k+1)}(x):=e(e^{(k)}(x))$. Introduce the generating functions 
$$\sum_{n\geq0}\tilde{B}_n^{(k)}\frac{x^n}{n!}=1-e^{(k)}(x)$$
and call $\tilde{B}_n^{(k)}$ super complementary Bell numbers. Experimental evidence prompts:

Question. For each odd integer $k$, the numbers $\tilde{B}_n^{(k)}=0$ iff $n=2$. Is this true?  

Note. $e^{(1)}(x)=1-e^{1-e^x}$ and hence $\tilde{B}_n^{(1)}=\tilde{B}_n$.
Examples. $e^{(2)}(x)=1-e^{1-e^{1-e^x}}$ and $e^{(3)}(x)=1-e^{1-e^{1-e^{1-e^x}}}$. The first few values of $\tilde{B}_n^{(3)}$:
$1, -1, 0, 2, 2, -14, -53, 129, 1668, 1607, \dots$.

REPLY [2 votes]: Just a comment, not an answer. 
It might be helpful to get an idea when one can see, how the coefficients of the odd iterates can be expressed as polynomials on the iteration-index. Here is the table of coefficients of the exponential generating functions of the first 8 odd iterates (row 1: Uppuluri-Carpenter-numbers, row 2: numbers as given in the example in the OP for the 3'rd iterate) with the iteration index $h$:
   h | j | x^0 x  x^2  x^3  x^4   x^5    x^6    x^7     x^8 ...
  ---+---+-----------------------------------------------------
   1 | 1 | 1  -1    0    1    1    -2     -9     -9      50 ...
   3 | 2 | 1  -1    0    2    2   -14    -53    129    1668
   5 | 3 | 1  -1    0    3    3   -36   -132    764    7514
   7 | 4 | 1  -1    0    4    4   -68   -246   2246   20248
   9 | 5 | 1  -1    0    5    5  -110   -395   4925   42530
  11 | 6 | 1  -1    0    6    6  -162   -579   9151   77020
  13 | 7 | 1  -1    0    7    7  -224   -798  15274  126378
  15 | 8 | 1  -1    0    8    8  -296  -1052  23644  193264
  ...|... ... 

The entries along the columns can be calculated by polynomials depending on $h$ or even better on the consecutive rowindex $j$. The generating polynomials for the columns, in terms of $h$, are
 coef 
  at : polynomial in h
 -----------------------  
 x^0 :   1
 x^1 :  -1
 x^2 :   0
 x^3 :  1/2  +  1/2*h 
 x^4 :  1/2  +  1/2*h 
 x^5 :  1/4  -   1 *h  -  5/4*h^2
 x^6 : -1/8  -  9/2*h  - 35/8*h^2
 x^7 :  -1   -283/24*h -  7/2*h^2  + 175/24*h^3
 x^8 :-19/4  -365/12*h +119/4*h^2  + 665/12*h^3
 ... : ....

and in terms of the rowindex $j$ are      
 coeff
  at : polynomial in j
 -----------------------  
 x^0 :  1
 x^1 : -1
 x^2 :  0
 x^3 :  0 +    1 *j
 x^4 :  0 +    1 *j
 x^5 :  0 +    3 *j  -     5 *j^2 
 x^6 :  0 +  17/2*j  -   35/2*j^2 
 x^7 :  0 + 205/6*j  -  203/2*j^2  +  175/3*j^3
 x^8 :  0 + 458/3*j  -   546 *j^2  + 1330/3*j^3
 ... : ...


A piece of code in Pari/GP to reproduce the first table is
list=matrix(12,16)
tmp =  exp(x);
tmp =  exp(1-tmp); \\ first iterate
list[1,] = polcoeffs(serlaplace(tmp),16); \\ "polcoeffs" - user function 
            \\ gives the first few coefficients as a vector (here of length 16)
{ for(k=2,12,
        tmp =  exp(1-tmp);tmp =  exp(1-tmp); \\ use the odd iterations only
        list[k,]=polcoeffs(serlaplace(tmp),16)
     ); }
 printp (list  )


[Update] 
It might also be of interest, that the iterates of the Uppuluri-Carpenter-numbers can be created using the matrix-exponential of the (lower triangular) Pascalmatrix where the diagonal is removed. That removal of the diagonal makes the exponential-series for the matrix-exponential a finite sum for any truncation of size (which might in turn be of help for the finding of the proof of the conjecture) .         
Let P be the lower triangular Pascalmatrix truncated to some size $n \times n$, and $I$ the identity matrix of same size.
Then the iterates can be computed by:
$$ U_1 = \operatorname{Exp}( I - P)  \\
U_2 = \operatorname{Exp}( I - U_1)  \\
U_3 = \operatorname{Exp}( I - U_2)  \\
...
$$
The numbers of interest occur in the first column of the odd-indexed iterates $U_1,U_3,U_5,...$.
Here are top-left snippets of $U_1,U_3,U_5$: 
   U_1                                  U_3                       U_5
   -----------------------------+------------------------+-------------------
   1     .     .      .      .  |       1      .      .  |        1        .       .  |
  -1     1     .      .      .  |      -1      1      .  |       -1        1       .  |
   0    -2     1      .      .  |       0     -2      1  |        0       -2       1  |
   1     0    -3      1      .  |       2      0     -3  |        3        0      -3  |
   1     4     0     -4      1  |       2      8      0  |        3       12       0  |
  -2     5    10      0     -5  |     -14     10     20  |      -36       15      30  |
  -9   -12    15     20      0  |     -53    -84     30  |     -132     -216      45  |
  -9   -63   -42     35     35  |     129   -371   -294  |      764     -924    -756  |
  50   -72  -252   -112     70  |    1668   1032  -1484  |     7514     6112   -3696  |
 267   450  -324   -756   -252  |    1607  15012   4644  |   -14838    67626   27504  |
 413  2670  2250  -1080  -1890  |  -58515  16070  75060  |  -547854  -148380  338130  |
   -     -     -      -      -  +       -      -      -  +        -        -       -  +