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A229233 O.g.f.: Sum_{n>=0} x^n / Product_{k=1..n} (1 - n*k*x).

%I A229233 #17 Jan 30 2022 11:39:24
%S A229233 1,1,2,8,48,387,4043,52425,819346,15133184,324769270,7986143453,
%T A229233 222514878501,6958782341565,242274294115558,9324382604206368,
%U A229233 394282071192289024,18218582054356563951,915480348188869318723,49812603754178905560085,2923492374797360684715882
%N A229233 O.g.f.: Sum_{n>=0} x^n / Product_{k=1..n} (1 - n*k*x).
%C A229233 Compare to an o.g.f. of Bell numbers (A000110): Sum_{n>=0} x^n/Product_{k=1..n} (1-k*x).
%H A229233 Seiichi Manyama, <a href="/A229233/b229233.txt">Table of n, a(n) for n = 0..332</a>
%F A229233 a(n) = Sum_{k=0..n} k^(n-k) * Stirling2(n, k).
%F A229233 E.g.f.: Sum_{n>=0} (exp(n*x) - 1)^n / (n! * n^n).
%e A229233 O.g.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 48*x^4 + 387*x^5 + 4043*x^6 +...
%e A229233 where
%e A229233 A(x) = 1 + x/(1-x) + x^2/((1-2*1*x)*(1-2*2*x)) + x^3/((1-3*1*x)*(1-3*2*x)*(1-3*3*x)) + x^4/((1-4*1*x)*(1-4*2*x)*(1-4*3*x)*(1-4*4*x)) +...
%e A229233 Exponential Generating Function.
%e A229233 E.g.f.: E(x) = 1 + x + 2*x^2/2! + 8*x^3/3! + 48*x^4/4! + 387*x^5/5! +...
%e A229233 where
%e A229233 E(x) = 1 + (exp(x)-1) + (exp(2*x)-1)^2/(2!*2^2) + (exp(3*x)-1)^3/(3!*3^3) + (exp(4*x)-1)^4/(4!*4^4) + (exp(5*x)-1)^5/(5!*5^5) +...
%t A229233 Flatten[{1,Table[Sum[k^(n-k) * StirlingS2[n, k],{k,0,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, May 08 2014 *)
%o A229233 (PARI) {a(n)=polcoeff(sum(m=0,n,x^m/prod(k=1,m,1-m*k*x +x*O(x^n))),n)}
%o A229233 for(n=0,30,print1(a(n),", "))
%o A229233 (PARI) {a(n)=n!*polcoeff(sum(m=0,n,(exp(m*x+x*O(x^n))-1)^m/(m!*m^m)),n)}
%o A229233 for(n=0,30,print1(a(n),", "))
%o A229233 (PARI) {a(n)=sum(k=0, n, k^(n-k) * stirling(n, k, 2))}
%o A229233 for(n=0,30,print1(a(n),", "))
%Y A229233 Cf. A229234, A220181, A108459, A122399.
%Y A229233 Cf. A229258, A229259, A229260, A229261.
%K A229233 nonn
%O A229233 0,3
%A A229233 _Paul D. Hanna_, Sep 17 2013