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A193931 E.g.f. A(x) = exp(x+x^2+x^3+x^4+x^5).

%I A193931 #13 Sep 29 2017 12:01:52
%S A193931 1,1,3,13,73,501,3331,27553,253233,2540233,27340291,309811701,
%T A193931 3843476473,50560182973,701098196163,10205952248521,155809733115361,
%U A193931 2506135027165713,42013633806350083,732584456250306013,13270900741926553641,249625454707702681861
%N A193931 E.g.f. A(x) = exp(x+x^2+x^3+x^4+x^5).
%H A193931 Alois P. Heinz, <a href="/A193931/b193931.txt">Table of n, a(n) for n = 0..500</a>
%F A193931 a(n)=n!*sum(k=1..n, sum(i=0..(n-k)/5, (-1)^i*binomial(k,k-i)*binomial(n-5*i-1,k-1))/k!), n>0, a(0)=1.
%F A193931 E.g.f.: 1 + x*(E(0)-1)/(x+1) where E(k) =  1 + (1+x+x^2+x^3+x^4)/(k+1)/(1-x/(x+1/E(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Jan 27 2013
%p A193931 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A193931       a(n-j)*binomial(n-1, j-1)*j!, j=1..min(n, 5)))
%p A193931     end:
%p A193931 seq(a(n), n=0..23);  # _Alois P. Heinz_, Sep 29 2017
%o A193931 (Maxima)
%o A193931 a(n):=if n=0 then 1 else n!*sum(sum((-1)^i*binomial(k,k-i)*binomial(n-5*i-1,k-1),i,0,(n-k)/5)/k!,k,1,n);
%K A193931 nonn
%O A193931 0,3
%A A193931 _Vladimir Kruchinin_, Aug 09 2011