t(z):=sum(n↑(n-1)*z↑n/factorial(n),n,1,inf);
                                    N - 1  N
                                   N      Z
(D1)                   T(Z) := SUM(---------, N, 1, INF)
                                      N!

(C2) 
taylor(t(z),z,0,5);

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                                 3      4        5
                          2   3 Z    8 Z    125 Z
(D2)/T/              Z + Z  + ---- + ---- + ------ + . . .
                               2      3       24

(C3) 
ratcoeff(d2,z↑3);
                                       3
(D3)/R/                                -
                                       2

(C4) 
taylor((t(z)↑2-t(z)↑3/2)/(1-t(z))↑(3/2),z,0,5);
                                      4        5
                       2      3   65 Z    259 Z
(D4)/T/               Z  + 3 Z  + ----- + ------ + . . .
                                    8       12

(C5) 
taylor(t(z)↑2/(1-t(z))↑(1/2),z,0,5);
                              3       4        5
                       2   5 Z    47 Z    667 Z
(D5)/T/               Z  + ---- + ----- + ------ + . . .
                            2       8       48

(C6) 
int(z,z,0,a);
(D6)                            INT(Z, Z, 0, A)

(C7) 
integrate(z,z,0,a);

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                                       2
                                      A
(D7)                                  --
                                      2

(C8) 
d3;
                                       3
(D8)/R/                                -
                                       2

(C9) 
d4;
                                      4        5
                       2      3   65 Z    259 Z
(D9)/T/               Z  + 3 Z  + ----- + ------ + . . .
                                    8       12

(C10) 
integrate(taylor(d4/t(z)/z,z,0,5),z,0,z);
                             4        3       2
                        271 Z  + 148 Z  + 96 Z  + 96 Z
(D10)                   ------------------------------
                                      96

(C11) 
taylor(%*t(z)/(1-t(z)),z,0,4);
                                           4
                           2      3   193 Z
(D11)/T/                  Z  + 3 Z  + ------ + . . .
                                        24

(C12) 
diff(t*(1-t)↑(1/2),t);
                                              T
(D12)                     SQRT(1 - T) - -------------
                                        2 SQRT(1 - T)

(C13) 
ratsimp(%*(1-t)↑(1/2));
                                     3 T - 2
(D13)                              - -------
                                        2

(C14)