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, 1, 2, 12, 181, 5237, 245776, 16954562, 1612833457, 202233823341, 32315380158578, 6409484794915012, 1544967825490593319, 444799853104579872759, 150750913498484630903772, 59410000121654748323276898, 26938215605761889373324449091, 13925028099872858626544313312207
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 2*x^2 + 12*x^3 + 181*x^4 + 5237*x^5 + 245776*x^6 +... where the logarithmic derivative is given by the series: 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)) +... Explicitly, 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 +... compare to: 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! +... where G(x) = (1-exp(-x)) + (1-exp(-2*x))^2/2 + (1-exp(-3*x))^3/3 + (1-exp(-4*x))^4/4 +...
Programs
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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)} for(n=0, 20, print1(a(n), ", ")) -
PARI
/* From g.f. exp( Sum_{n>=1} A092552(n)*x^n/n ): */ {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)} {A092552(n)=if(n<=0, 0, sum(k=1, n, k!*(k-1)! * Stirling2(n, k)^2))} {a(n)=polcoeff(exp(sum(m=1,n,A092552(m)*x^m/m) +x*O(x^n)),n)} for(n=0,20,print1(a(n),", "))