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A243440 O.g.f.: exp( Integral Sum_{n>=1} n! * n^(n-1) * x^(n-1) / Product_{k=1..n} (1 - k*x) dx ). 1

%I

%S 1,1,3,25,499,18897,1158175,104287909,12948389505,2119204222647,

%T 442024984454145,114447363118335099,36014003359662761889,

%U 13536516384259740525435,5989775500211255393302197,3082008257212085146469317911,1824650971940959528920159955650,1231558332755627626667173051846452

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

%e G.f.: A(x) = 1 + x + 3*x^2 + 25*x^3 + 499*x^4 + 18897*x^5 + 1158175*x^6 +...

%e The logarithmic derivative equals the series:

%e A'(x)/A(x) = 1/(1-x) + 2!*2*x/((1-x)*(1-2*x)) + 3!*3^2*x^2/((1-x)*(1-2*x)*(1-3*x)) + 4!*4^3*x^3/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)) + 5!*5^4*x^4/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)) +...

%e Explicitly, the logarithm of the o.g.f. begins:

%e log(A(x)) = x + 5*x^2/2 + 67*x^3/3 + 1889*x^4/4 + 91771*x^5/5 + 6828545*x^6/6 + 721578187*x^7/7 + 102730470449*x^8/8 +...

%o (PARI) {a(n)=polcoeff(exp(intformal(sum(m=1, n+1, m!*m^(m-1)*x^(m-1)/prod(k=1, m, 1-k*x+x*O(x^n))))), n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A084784, A243435.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jun 05 2014

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Last modified September 26 23:42 EDT 2021. Contains 347673 sequences. (Running on oeis4.)