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%I #14 Feb 03 2018 14:35:57
%S 1,1,3,29,686,30552,2191262,230356646,33349943718,6359939775042,
%T 1545000640114242,465750550069828422,170603300462464687996,
%U 74630981535308266499848,38429419191031108995080412,23008323194727484508595195772,15848730592891024979096686043722,12445298391963001703710163766096546
%N O.g.f.: exp( Integral Sum_{n>=1} n! * n^(n-1) * x^(n-1) / Product_{k=1..n} (1 - k*n*x) dx ).
%e G.f.: A(x) = 1 + x + 3*x^2 + 29*x^3 + 686*x^4 + 30552*x^5 + 2191262*x^6 +...
%e The logarithmic derivative equals the series:
%e A'(x)/A(x) = 1/(1-x) + 2!*2*x/((1-2*x)*(1-4*x)) + 3!*3^2*x^2/((1-3*x)*(1-6*x)*(1-9*x)) + 4!*4^3*x^3/((1-4*x)*(1-8*x)*(1-12*x)*(1-16*x)) + 5!*5^4*x^4/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)) +...
%e Explicitly, the logarithm of the o.g.f. begins:
%e log(A(x)) = x + 5*x^2/2 + 79*x^3/3 + 2621*x^4/4 + 149071*x^5/5 + 12954365*x^6/6 + 1596620719*x^7/7 + 264914218301*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-m*k*x+x*O(x^n))))), n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A084784, A243440.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 05 2014
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