%I M3976 N1645 #24 Oct 14 2023 15:39:54
%S 1,5,35,225,67375,853125,955040625,1861234375,151365980390625,
%T 142468185234375,10686017754521484375,8684623124912109375,
%U 5398544111530990341796875,54231540104196533203125,1161721704933873029968505859375
%N Denominators of coefficients in an asymptotic expansion of the confluent hypergeometric function F(1-b; 2; 4b).
%C Value of a(5) is incorrectly given as 66693 in Henrici paper. - _Sean A. Irvine_, Jun 20 2013
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Peter Henrici, <a href="https://doi.org/10.1145/320815.320819">Automatic computations with power series</a>, J. Assoc. Comput. Mach. 3 (1956), 10-15.
%F Let f(x) = [Sum_{k>=1}(3/(2*k+1)) * x^(2*k+1)]^(1/3) = x + (1/5)*x^3 + (18/175) * x^5 + ...; let g(x) be the Lagrange inversion of f(x), g(x) = REVERT(f(x)) = 1 - (1/5) * x^3 + (3/175) * x^5 + .... Then a(n) = denominator((2 * n + 1) * coeff(g(x), 2*n+1)). - _Sean A. Irvine_, Jun 20 2013
%t nmax = 14;
%t S = Sum[(3/(2k+1)) x^(2k+1), {k, 1, Infinity}]^(1/3) + O[x]^(3nmax) // Normal // Simplify[#, x > 0]& // InverseSeries[# + O[x]^(3nmax), x]&;
%t a[n_] := Denominator[(2n+1) SeriesCoefficient[S, {x, 0, 2n+1}]];
%t a /@ Range[0, nmax] (* _Jean-François Alcover_, Oct 01 2020 *)
%Y Cf. A002073 (numerators).
%K nonn,frac
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Sean A. Irvine_, Jun 20 2013