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A110083 a(n+1) = Sum_{k=0..n} (n!/k!)*binomial(n,k)*a(k).

%I A110083 #12 Jul 18 2025 08:31:22
%S A110083 1,1,2,8,50,442,5212,78664,1472756,33378740,898227944,28253387104,
%T A110083 1025373023848,42467845178632,1988513519453360,104413376937507488,
%U A110083 6104596110052561808,394921638012548722576,28112685278602155590944,2191142414957886078590080
%N A110083 a(n+1) = Sum_{k=0..n} (n!/k!)*binomial(n,k)*a(k).
%H A110083 Alois P. Heinz, <a href="/A110083/b110083.txt">Table of n, a(n) for n = 0..307</a>
%F A110083 Conjecture: a(n) = R(n,0) where R(n,k) = R(n-1,n-1) + n * Sum_{j=0..k-1} R(n-1,j) for 0 <= k <= n with R(0,0) = 1. - _Mikhail Kurkov_, Jul 18 2025
%p A110083 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A110083       a(n-i)*binomial(n-1, i-1)^2*(i-1)!, i=1..n))
%p A110083     end:
%p A110083 seq(a(n), n=0..20);  # _Alois P. Heinz_, Aug 13 2019
%t A110083 nmax=20; b = ConstantArray[0,nmax+2]; b[[1]]=1; Do[b[[n+2]] = Sum[n!/k!*Binomial[n,k]*b[[k+1]],{k,0,n}],{n,0,nmax}]; b (* _Vaclav Kotesovec_, Mar 02 2014 *)
%Y A110083 Cf. A001063.
%K A110083 easy,nonn
%O A110083 0,3
%A A110083 _Vladeta Jovovic_, Sep 04 2005