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EXAMPLE
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E.g.f.: A(x) = 1 + x + 31*x^2/2! + 117715*x^3/3! + 103060086811*x^4/4! +...
where
A(x) = 1 + (1-exp(-x)) + (1-exp(-2^2*x))^2 + (1-exp(-3^3*x))^3 + (1-exp(-4^4*x))^4 + (1-exp(-5^5*x))^5 + (1-exp(-6^6*x))^6 +...
Ordinary generating function.
O.g.f.: F(x) = 1 + x + 31*x^2 + 117715*x^3 + 103060086811*x^4 +...
where
F(x) = 1 + x/(1+x) + 2^4*2!*x^2/((1+2^2*1*x)*(1+2^2*2*x)) + 3^9*3!*x^3/((1+3^3*1*x)*(1+3^3*2*x)*(1+3^3*3*x)) + 4^16*4!*x^4/((1+4^4*1*x)*(1+4^4*2*x)*(1+4^4*3*x)*(1+4^4*4*x)) + 5^25*5!*x^5/((1+5^5*1*x)*(1+5^5*2*x)*(1+5^5*3*x)*(1+5^5*4*x)*(1+5^5*5*x)) +...
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MATHEMATICA
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Flatten[{1, Table[Sum[(-1)^(n-k)*(k^k)^n*k!*StirlingS2[n, k], {k, 1, n}], {n, 1, 10}]}] (* Vaclav Kotesovec, Aug 24 2017 *)
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PROG
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(PARI) {a(n)=n!*polcoeff(sum(k=0, n, (1-exp(-k^k*x +x*O(x^n)))^k), n)}
for(n=0, 15, print1(a(n), ", "))
(PARI) {a(n)=polcoeff(sum(m=0, n, m^(m^2)*m!*x^m/prod(k=1, m, 1+m^m*k*x +x*O(x^n))), n)}
for(n=0, 15, print1(a(n), ", "))
(PARI) {a(n) = sum(k=0, n, (-1)^(n-k)*k^(k*n)*k!*stirling(n, k, 2))}
for(n=0, 10, print1(a(n), ", "))
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