A071787 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^[...]]]].

1, 1, 1, 2, 5, 12, 47, 120, 12917, 33840, 329458703, 874222560, 4526064144016687091, 12096849691539466560, 4339254722819592663241773932837977109

Offset

1,4

Comments

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?

Links

Table of n, a(n) for n=1..15.

Example

1/(1-x) = e^[(1/1)*x*e^[(1/2)*x*e^[(5/12)*x*e^[(47/120)*x*e^[...]]]]

Mathematica

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 *)

Crossrefs

Cf. A068985 (1/e).

Sequence in context: A172239 A183758 A334811 * A343813 A332791 A291484

Adjacent sequences: A071784 A071785 A071786 * A071788 A071789 A071790

Keyword

nonn

Author

Paul D. Hanna, Jun 06 2002

Extensions

Terms from a(11) through a(16) were supplied by David W. Cantrell (DWCantrell(AT)sigmaxi.net)

Status

approved