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%I #3 Jun 11 2014 01:42:17
%S 1,1,4,42,909,33969,1948514,158770640,17419561466,2474812055850,
%T 441910422152592,96867456432497772,25572186966552515130,
%U 8002470602289313981938,2929213328377453597523820,1239933908245021946285387592,601020049737946926794959094457
%N G.f.: exp( Integral Sum_{n>=1} n!*n^n*x^(n-1) / Product_{k=1..n} (1+k*n*x) dx ).
%F G.f.: exp( Sum_{n>=1} A220181(n)*x^n/n ) where Sum_{n>=1} A220181(n)*x^n/n! = Sum_{n>=1} (1 - exp(-n*x))^n.
%e G.f.: A(x) = 1 + x + 4*x^2 + 42*x^3 + 909*x^4 + 33969*x^5 + 1948514*x^6 +...
%e where the logarithmic derivative is given by the series:
%e A'(x)/A(x) = 1/(1+x) + 2!*2^2*x/((1+1*2*x)*(1+2*2*x)) + 3!*3^3*x^2/((1+1*3*x)*(1+2*3*x)*(1+3*3*x)) + 4!*4^4*x^3/((1+1*4*x)*(1+2*4*x)*(1+3*4*x)*(1+4*4*x)) + 5!*5^5*x^4/((1+1*5*x)*(1+2*5*x)*(1+3*5*x)*(1+4*5*x)*(1+5*5*x)) +...
%e Explicitly,
%e A'(x)/A(x) = 1 + 7*x + 115*x^2 + 3451*x^3 + 164731*x^4 + 11467387*x^5 +...+ A220181(n+1)*x^n +...
%e compare to:
%e G(x) = x + 7*x^2/2! + 115*x^3/3! + 3451*x^4/4! + 164731*x^5/5! + 11467387*x^6/6! +...+ A220181(n)*x^n/n! +...
%e where G(x) = (1-exp(-x)) + (1-exp(-2*x))^2 + (1-exp(-3*x))^3 + (1-exp(-4*x))^4 +...
%o (PARI) {a(n)=local(A=1+x); A=exp(intformal(sum(m=1, n+1, m^m*m!*x^(m-1)/prod(k=1, m, 1+m*k*x +x*O(x^n))))); polcoeff(A,n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) /* From g.f. exp( Sum_{n>=1} A220181(n)*x^n/n ): */
%o {A220181(n)=n!*polcoeff(sum(k=0, n, (1-exp(-k*x+x*O(x^n)))^k), n)}
%o {a(n)=polcoeff(exp(sum(m=1,n,A220181(m)*x^m/m) +x*O(x^n)),n)}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A220181, A243807.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 11 2014
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