FindSequenceFunction (modulo some fondling) gives closed forms for A153229 <https://oeis.org/A153229>, A153229 <https://oeis.org/A153229>, and A005165 <https://oeis.org/A005165>: In[317]:= Table[ FullSimplify[-(-1)^n E ExpIntegralEi[-1] + E ExpIntegralE[n, 1] Gamma[n]], {n, 9}] Out[317]= {0, 1, 0, 2, 4, 20, 100, 620, 4420} But strangely, In[313]:= FullSimplify[(-1)^ n E (-(-1)^n ExpIntegralEi[-1] + ExpIntegralE[n, 1] Gamma[n]), n \[Element] Integers && n > 0] Out[313]= (-1)^n E (-(-1)^n ExpIntegralEi[-1] + ExpIntegralE[n, 1] Gamma[n]) instead of In[318]:= Factor[ FullSimplify[#, n \[Element] Integers && n > 0] & /@ Expand[%313]] Out[318]= -E (ExpIntegralEi[-1] - (-1)^n ExpIntegralE[n, 1] Gamma[n]) which it concedes is two LeafCounts smaller. --rwg <https://oeis.org/A005165>