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A243807 G.f.: exp( Integral Sum_{n>=1} n!*n^(n-1)*x^(n-1) / Product_{k=1..n} (1+k*n*x) dx ). 1

%I

%S 1,1,2,12,181,5237,245776,16954562,1612833457,202233823341,

%T 32315380158578,6409484794915012,1544967825490593319,

%U 444799853104579872759,150750913498484630903772,59410000121654748323276898,26938215605761889373324449091,13925028099872858626544313312207

%N G.f.: exp( Integral Sum_{n>=1} n!*n^(n-1)*x^(n-1) / Product_{k=1..n} (1+k*n*x) dx ).

%F G.f.: exp( Sum_{n>=1} A092552(n)*x^n/n ) where Sum_{n>=1} A092552(n)*x^n/n! = Sum_{n>=1} (1 - exp(-n*x))^n / n.

%e G.f.: A(x) = 1 + x + 2*x^2 + 12*x^3 + 181*x^4 + 5237*x^5 + 245776*x^6 +...

%e where the logarithmic derivative is given by the series:

%e A'(x)/A(x) = 1/(1+x) + 2!*2^1*x/((1+1*2*x)*(1+2*2*x)) + 3!*3^2*x^2/((1+1*3*x)*(1+2*3*x)*(1+3*3*x)) + 4!*4^3*x^3/((1+1*4*x)*(1+2*4*x)*(1+3*4*x)*(1+4*4*x)) + 5!*5^4*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 + 3*x + 31*x^2 + 675*x^3 + 25231*x^4 + 1441923*x^5 + 116914351*x^6 +...+ A092552(n+1)*x^n +...

%e compare to:

%e G(x) = x + 3*x^2/2! + 31*x^3/3! + 675*x^4/4! + 25231*x^5/5! + 1441923*x^6/6! +...+ A092552(n)*x^n/n! +...

%e where G(x) = (1-exp(-x)) + (1-exp(-2*x))^2/2 + (1-exp(-3*x))^3/3 + (1-exp(-4*x))^4/4 +...

%o (PARI) {a(n)=local(A=1+x); A=exp(intformal(sum(m=1, n+1, m^(m-1)*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} A092552(n)*x^n/n ): */

%o {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}

%o {A092552(n)=if(n<=0, 0, sum(k=1, n, k!*(k-1)! * Stirling2(n, k)^2))}

%o {a(n)=polcoeff(exp(sum(m=1,n,A092552(m)*x^m/m) +x*O(x^n)),n)}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A092552, A243809.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jun 11 2014

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Last modified March 28 05:38 EDT 2020. Contains 333073 sequences. (Running on oeis4.)