A002074 Denominators of coefficients in an asymptotic expansion of the confluent hypergeometric function F(1-b; 2; 4b). (Formerly M3976 N1645)

1, 5, 35, 225, 67375, 853125, 955040625, 1861234375, 151365980390625, 142468185234375, 10686017754521484375, 8684623124912109375, 5398544111530990341796875, 54231540104196533203125, 1161721704933873029968505859375

Offset

0,2

Comments

Value of a(5) is incorrectly given as 66693 in Henrici paper. - Sean A. Irvine, Jun 20 2013

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=0..14.

Peter Henrici, Automatic computations with power series, J. Assoc. Comput. Mach. 3 (1956), 10-15.

Formula

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

Mathematica

nmax = 14;
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]&;
a[n_] := Denominator[(2n+1) SeriesCoefficient[S, {x, 0, 2n+1}]];
a /@ Range[0, nmax] (* Jean-François Alcover, Oct 01 2020 *)

Crossrefs

Cf. A002073 (numerators).

Sequence in context: A320071 A024062 A180250 * A187444 A166176 A154780

Adjacent sequences: A002071 A002072 A002073 * A002075 A002076 A002077

Keyword

nonn,frac

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, Jun 20 2013

Status

approved