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

%I A005146 M5308 #33 May 08 2018 15:11:54
%S A005146 1,1,53,195,22999,29944523,109535241009,29404527905795295658,
%T A005146 455377030420113432210116914702,
%U A005146 26370812569397719001931992945645578779849,152537496709054809881638897472985990866753853122697839,100043420063777451042472529806266909090824649341814868347109676190691
%N A005146 Numerators of numbers occurring in continued fraction connected with expansion of gamma function.
%D A005146 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 258.
%D A005146 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A005146 H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 365.
%H A005146 Vincenzo Librandi, <a href="/A005146/b005146.txt">Table of n, a(n) for n = 0..30</a>
%H A005146 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A005146 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 258.
%H A005146 B. W. Char, <a href="http://www.jstor.org/stable/2006103">On Stieltjes' continued fraction for the gamma function</a>, Math. Comp., 34 (1980), 547-551.
%H A005146 Peter Luschny, <a href="/A005146/a005146.txt">Maple program for A005146/A005147</a>
%H A005146 Peter Luschny, <a href="http://www.luschny.de/math/factorial/approx/continuedfraction.html">Continued fraction</a>
%H A005146 W. Wang, <a href="http://dx.doi.org/10.1016/j.jnt.2015.12.016">Unified approaches to the approximations of the gamma function</a>, J. Number Theory (2016).
%t A005146 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] =
%t A005146 If[OddQ[k], m[n+1, k-2]+m[n+1, k-1]-m[n, k-1],
%t A005146 m[n+1, k-2]*m[n+1, k-1]/m[n, k-1]], {k, 3, len}, {n, 0,
%t A005146 len-k+1}]; Do[m[n, 1] = s[n], {n, 0, len}];
%t A005146 Table[m[0, k], {k, 1, len}] // Numerator
%t A005146 (* _Jean-François Alcover_, May 24 2011, after _Peter Luschny_ *)
%Y A005146 Cf. A005147.
%K A005146 nonn,frac,nice
%O A005146 0,3
%A A005146 _Simon Plouffe_ and _N. J. A. Sloane_
%E A005146 More terms from _Rainer Rosenthal_, Jan 11 2007