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%I #14 Dec 25 2021 07:13:36
%S 1,1,1,2,5,12,47,120,12917,33840,329458703,874222560,
%T 4526064144016687091,12096849691539466560,
%U 4339254722819592663241773932837977109
%N Continued exponent expansion of the power series 1/(1-x); odd terms being numerators and even terms being denominators of the rational terms of the expansion: 1/(1-x) = e^[(a(1)/a(2))*x*e^[(a(3)/a(4))*x*e^[(a(5)/a(6))*x*e^[...]]]].
%C The fractions a(2n-1)/a(2n) form a monotonically decreasing sequence with the limit being 1/e = 0.3678794411714.... What is the rate of growth of the terms?
%e 1/(1-x) = e^[(1/1)*x*e^[(1/2)*x*e^[(5/12)*x*e^[(47/120)*x*e^[...]]]]
%t a[0, _] = 1; a[n_, m_:0] := a[n, m] = (a[n-1, m+1] - Sum[k a[n, k-1] a[n-1, m-k+1], {k, m}]/(m+1))/a[n-1, 0]; Table[NumeratorDenominator[a[n]], {n, 10}] // Flatten (* _Vladimir Reshetnikov_, Dec 23 2021 *)
%Y Cf. A068985 (1/e).
%K nonn
%O 1,4
%A _Paul D. Hanna_, Jun 06 2002
%E Terms from a(11) through a(16) were supplied by David W. Cantrell (DWCantrell(AT)sigmaxi.net)
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