A002073 Numerators of coefficients in an asymptotic expansion of the confluent hypergeometric function F(1-b; 2; 4b). (Formerly M2268 N0897)

1, -3, 3, 2, -48, -362, -49711, 13952, 574406627, 64140842, -841796802304, -326397876886, -23544490420768844, 45123679545344, 449339765798227104271, 17766371321955738181048, -20395677580116057792512, -74026374065532274752108118

Offset

0,2

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..17.

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) = numerator((2 * n + 1) * coeff(g(x), 2*n+1)). - Sean A. Irvine, Jun 20 2013

Mathematica

nmax = 17;
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_] := Numerator[(2n+1) SeriesCoefficient[S, {x, 0, 2n+1}]];
a /@ Range[0, nmax] (* Jean-François Alcover, Oct 01 2020 *)

Crossrefs

Cf. A002074 (denominators).

Sequence in context: A196544 A289893 A265466 * A247093 A329273 A130719

Adjacent sequences: A002070 A002071 A002072 * A002074 A002075 A002076

Keyword

sign

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, Jun 20 2013

Status

approved