This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A019727 #94 Feb 16 2025 08:32:33
%S A019727 2,5,0,6,6,2,8,2,7,4,6,3,1,0,0,0,5,0,2,4,1,5,7,6,5,2,8,4,8,1,1,0,4,5,
%T A019727 2,5,3,0,0,6,9,8,6,7,4,0,6,0,9,9,3,8,3,1,6,6,2,9,9,2,3,5,7,6,3,4,2,2,
%U A019727 9,3,6,5,4,6,0,7,8,4,1,9,7,4,9,4,6,5,9,5,8,3,8,3,7,8,0,5,7,2,6
%N A019727 Decimal expansion of sqrt(2*Pi).
%C A019727 Pickover says that the expression: lim_{n->oo} e^n(n!) / (n^n * sqrt(n)) = sqrt(2*Pi) is beautiful because it connects Pi, e, radicals, factorials and infinite limits. - _Jason Earls_, Mar 16 2001
%C A019727 Appears in the formula of the normal distribution. - _Johannes W. Meijer_, Feb 23 2013
%C A019727 sqrt(2*Pi)*sqrt(n) is the expected height of a labeled random tree of order n (see Rényi, Szekeres, 1967, formula (4.6)). - _Hugo Pfoertner_, May 18 2023
%C A019727 The constant in the formula known as "Stirling's approximation" (or "Stirling's formula"). It is sometimes called Stirling constant. The formula without the exact value of the constant was discovered by the French mathematician Abraham de Moivre (1667-1754), and was published in his book (1730). The exact value of the constant was found by the Scottish mathematician James Stirling (1692-1770) and was published in his book "Methodus differentialis" (1730). - _Amiram Eldar_, Jul 08 2023
%D A019727 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 137.
%D A019727 Clifford A. Pickover, Wonders of Numbers, Oxford University Press, NY, 2001, p. 307.
%D A019727 David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 45.
%H A019727 Harry J. Smith, <a href="/A019727/b019727.txt">Table of n, a(n) for n = 1..20000</a>
%H A019727 Mohammad K. Azarian, <a href="http://www.jstor.org/stable/27646572">An Expression for Pi, Problem #870</a>, College Mathematics Journal, Vol. 39, No. 1, January 2008, p. 66. <a href="http://www.jstor.org/stable/27646723">Solution</a> appeared in Vol. 40, No. 1, January 2009, pp. 62-64.
%H A019727 Abraham de Moivre, <a href="https://archive.org/details/bub_gb_TFX1165yEc4C">Miscellanea Analytica de Seriebus et Quadraturis</a>, London, England: J. Tonson & J. Watts, 1730, pp. 96-106.
%H A019727 K. Kimoto, N. Kurokawa, C. Sonoki, and M. Wakayama, <a href="https://doi.org/10.2996/kmj/1104247354">Some examples of generalized zeta regularized products</a>, Kodai Math. J. 27 (2004), 321-335.
%H A019727 Clifford A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," <a href="https://zbmath.org/?q=an:0983.00008">Zentralblatt review</a>.
%H A019727 A. Rényi and G. Szekeres, <a href="https://doi.org/10.1017/S1446788700004432">On the height of trees</a>, Journal of the Australian Mathematical Society , Volume 7 , Issue 4 , November 1967 , pp. 497-507.
%H A019727 James Stirling, <a href="https://archive.org/details/bub_gb_71ZHAAAAYAAJ/page/n145/mode/2up">Methodus differentialis, sive tractatus de summatione et interpolatione serierum infinitarum</a>, London, 1730. See Propositio XXVIII, pp. 135-139.
%H A019727 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NormalDistribution.html">Normal Distribution</a>.
%H A019727 Wikipedia, <a href="https://en.wikipedia.org/wiki/Stirling%27s_approximation">Stirling's approximation</a>.
%H A019727 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A019727 Equals lim_{n->oo} e^n*(n!)/n^n*sqrt(n).
%F A019727 Also equals Integral_{x >= 0} W(1/x^2) where W is the Lambert function, which is also known as ProductLog. - _Jean-François Alcover_, May 27 2013
%F A019727 Also equals the generalized Glaisher-Kinkelin constant A_0, see the Finch reference. - _Jean-François Alcover_, Dec 23 2014
%F A019727 Equals exp(-zeta'(0)). See Kimoto et al. - _Michel Marcus_, Jun 27 2019
%e A019727 2.506628274631000502415765284811045253006986740609938316629923576342293....
%t A019727 RealDigits[Sqrt[2Pi],10,120][[1]] (* _Harvey P. Dale_, Dec 12 2012 *)
%o A019727 (PARI) default(realprecision, 20080); x=sqrt(2*Pi); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b019727.txt", n, " ", d)); \\ _Harry J. Smith_, May 31 2009
%o A019727 (Maxima) fpprec: 100$ ev(bfloat(sqrt(2*%pi))); /* _Martin Ettl_, Oct 11 2012 */
%o A019727 (Magma) R:= RealField(100); Sqrt(2*Pi(R)); // _G. C. Greubel_, Mar 08 2018
%Y A019727 Cf. A058293 (continued fraction), A231863 (inverse), A000796 (Pi).
%K A019727 nonn,cons
%O A019727 1,1
%A A019727 _N. J. A. Sloane_