A005146 Numerators of numbers occurring in continued fraction connected with expansion of gamma function.

1, 1, 53, 195, 22999, 29944523, 109535241009, 29404527905795295658, 455377030420113432210116914702, 26370812569397719001931992945645578779849, 152537496709054809881638897472985990866753853122697839, 100043420063777451042472529806266909090824649341814868347109676190691

Offset: 0

Author

Simon Plouffe and N. J. A. Sloane

Keywords

• nonn
• frac
• nice

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 258.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 365.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..30
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 258.
• B. W. Char, On Stieltjes' continued fraction for the gamma function, Math. Comp., 34 (1980), 547-551.
• Peter Luschny, Maple program for A005146/A005147
• Peter Luschny, Continued fraction
• W. Wang, Unified approaches to the approximations of the gamma function, J. Number Theory (2016).

Crossrefs

Cf. A005147.

Programs

Mathematica

len = 12; s[p_] := (-1)^p * BernoulliB[2p+2]/(2p+1)/(2p+2); Do[m[n, 1] = 0, {n, 0, len}]; Do[m[n, 2] = s[n+1]/s[n], {n, 0, len-1}]; Do[m[n, k] =
If[OddQ[k], m[n+1, k-2]+m[n+1, k-1]-m[n, k-1],
m[n+1, k-2]*m[n+1, k-1]/m[n, k-1]], {k, 3, len}, {n, 0,
len-k+1}]; Do[m[n, 1] = s[n], {n, 0, len}];
Table[m[0, k], {k, 1, len}] // Numerator
(* Jean-François Alcover, May 24 2011, after Peter Luschny *)

Extensions

More terms from Rainer Rosenthal, Jan 11 2007