A193931 - OEIS

%I #13 Sep 29 2017 12:01:52

%S 1,1,3,13,73,501,3331,27553,253233,2540233,27340291,309811701,

%T 3843476473,50560182973,701098196163,10205952248521,155809733115361,

%U 2506135027165713,42013633806350083,732584456250306013,13270900741926553641,249625454707702681861

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

%H Alois P. Heinz, <a href="/A193931/b193931.txt">Table of n, a(n) for n = 0..500</a>

%F 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 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 a:= proc(n) option remember; `if`(n=0, 1, add(

%p       a(n-j)*binomial(n-1, j-1)*j!, j=1..min(n, 5)))

%p     end:

%p seq(a(n), n=0..23);  # _Alois P. Heinz_, Sep 29 2017

%o (Maxima)

%o 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 nonn

%O 0,3

%A _Vladimir Kruchinin_, Aug 09 2011