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A207214 E.g.f.: Sum_{n>=0} exp(n*x) * Product_{k=1..n} (exp(k*x) - 1).

%I A207214 #20 Feb 05 2020 23:52:10
%S A207214 1,1,7,85,1759,55621,2501407,151984645,12004046719,1196068161541,
%T A207214 146792747463007,21762540250822405,3834791755438306879,
%U A207214 792270319634586707461,189687840256042278859807,52103089179906338874671365,16275196750916467736633834239
%N A207214 E.g.f.: Sum_{n>=0} exp(n*x) * Product_{k=1..n} (exp(k*x) - 1).
%C A207214 Compare the e.g.f. to the identity:
%C A207214 exp(-x) = Sum_{n>=0} exp(n*x) * Product_{k=1..n} (1 - exp(k*x)).
%H A207214 Vaclav Kotesovec, <a href="/A207214/b207214.txt">Table of n, a(n) for n = 0..180</a>
%H A207214 Hsien-Kuei Hwang, Emma Yu Jin, <a href="https://arxiv.org/abs/1911.06690">Asymptotics and statistics on Fishburn matrices and their generalizations</a>, arXiv:1911.06690 [math.CO], 2019.
%F A207214 E.g.f. A(x) satisfies: A(x) = exp(-x)*(2*G(x) - 1),
%F A207214 where G(x) = Sum_{n>=0} Product_{k=1..n} (exp(k*x) - 1) = e.g.f. of A158690.
%F A207214 a(n) ~ sqrt(2) * 12^(n+1) * (n!)^2 / Pi^(2*n+2). - _Vaclav Kotesovec_, May 05 2014
%e A207214 E.g.f.: A(x) = 1 + x + 7*x^2/2! + 85*x^3/3! + 1759*x^4/4! + 55621*x^5/5! +...
%e A207214 such that, by definition,
%e A207214 A(x) = 1 + exp(x) * (exp(x)-1) + exp(2*x) * (exp(x)-1)*(exp(2*x)-1)
%e A207214 + exp(3*x) * (exp(x)-1)*(exp(2*x)-1)*(exp(3*x)-1)
%e A207214 + exp(4*x) * (exp(x)-1)*(exp(2*x)-1)*(exp(3*x)-1)*(exp(4*x)-1) +...
%e A207214 The related e.g.f. of A158690 equals the series:
%e A207214 G(x) = 1 + (exp(x)-1) + (exp(x)-1)*(exp(2*x)-1)
%e A207214 + (exp(x)-1)*(exp(2*x)-1)*(exp(3*x)-1)
%e A207214 + (exp(x)-1)*(exp(2*x)-1)*(exp(3*x)-1)*(exp(4*x)-1) +...
%e A207214 or, more explicitly,
%e A207214 G(x) = 1 + x + 5*x^2/2! + 55*x^3/3! + 1073*x^4/4! + 32671*x^5/5! +...
%e A207214 such that G(x) satisfies:
%e A207214 G(x) = (1 + exp(x)*A(x))/2.
%o A207214 (PARI) {a(n)=n!*polcoeff(sum(m=0,n+1,exp(m*x+x*O(x^n))*prod(k=1,m,exp(k*x+x*O(x^n))-1)),n)}
%o A207214 for(n=0,20,print1(a(n),", "))
%Y A207214 Cf. A158690.
%K A207214 nonn
%O A207214 0,3
%A A207214 _Paul D. Hanna_, Feb 16 2012