%I #18 Jan 30 2022 11:40:57
%S 1,1,3,19,189,2671,50253,1203679,35548509,1263153631,52973381853,
%T 2581493517439,144317666200029,9156299509121311,653254398215833053,
%U 51995430120141924799,4585316010326597014749,445304380297565009962591,47368550666889620425580253,5492643630110295899167573759
%N O.g.f.: Sum_{n>=0} n! * x^n / Product_{k=1..n} (1 - n*k*x).
%H Seiichi Manyama, <a href="/A229234/b229234.txt">Table of n, a(n) for n = 0..300</a>
%F a(n) = Sum_{k=0..n} k^(n-k) * k! * Stirling2(n, k).
%F E.g.f.: Sum_{n>=0} (exp(n*x) - 1)^n / n^n.
%e O.g.f.: A(x) = 1 + x + 3*x^2 + 19*x^3 + 189*x^4 + 2671*x^5 + 50253*x^6 +...
%e where
%e A(x) = 1 + x/(1-x) + 2!*x^2/((1-2*1*x)*(1-2*2*x)) + 3!*x^3/((1-3*1*x)*(1-3*2*x)*(1-3*3*x)) + 4!*x^4/((1-4*1*x)*(1-4*2*x)*(1-4*3*x)*(1-4*4*x)) +...
%e Exponential Generating Function.
%e E.g.f.: E(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 189*x^4/4! + 2671*x^5/5! +...
%e where
%e E(x) = 1 + (exp(x)-1) + (exp(2*x)-1)^2/2^2 + (exp(3*x)-1)^3/3^3 + (exp(4*x)-1)^4/4^4 + (exp(5*x)-1)^5/5^5 +...
%t Flatten[{1,Table[Sum[k^(n-k) * k! * StirlingS2[n, k],{k,0,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, May 08 2014 *)
%o (PARI) {a(n)=polcoeff(sum(m=0,n,m!*x^m/prod(k=1,m,1-m*k*x +x*O(x^n))),n)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) {a(n)=n!*polcoeff(sum(m=0,n,(exp(m*x+x*O(x^n))-1)^m/m^m),n)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) {a(n)=sum(k=0, n, k^(n-k) * k! * stirling(n, k, 2))}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A229233, A220181, A108459, A122399.
%Y Cf. A229258, A229259, A229260, A229261.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Sep 17 2013
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