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%I #5 Aug 31 2015 10:37:48
%S 1,2,0,5,4,1,5,1,5,1,4,0,2,9,8,3,1,5,4,8,3,1,4,1,1,3,7,5,7,8,4,4,8,8,
%T 0,1,2,0,7,2,7,0,4,1,9,1,8,8,2,2,4,9,5,8,1,0,9,3,2,7,1,8,2,3,5,4,4,7,
%U 6,4,8,8,1,0,6,5,5,1,1,2,5,5,6,3,2,1,7,0,3,6,5,2,2,9,3,7,9,8,9,0,8,0,6,7,9
%N Decimal expansion of -sm(-1), where sm(t) is the Dixonian elliptic function sm(t).
%C In the context of particle physics and in the case of a Yule branching process with two types of particles, this constant appears in the asymptotic expression of the probability that all particles be of the second type at time t as exp(-t)*smh(1).
%H Eric van Fossen Conrad and Philippe Flajolet, <a href="http://www.emis.de/journals/SLC/wpapers/s54conflaj.html">The Fermat cubic, elliptic functions, continued fractions and a combinatorial excursion</a>, Séminaire Lotharingien de Combinatoire 54 (2006), Article B54g, page 20.
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/WeierstrassEllipticFunction.html">Weierstrass Elliptic Function</a>
%e 1.205415151402983154831411375784488012072704191882249581...
%t sm[z_] := 6*WeierstrassP[z, {0, 1/27}]/(1 - 3*WeierstrassPPrime[z, {0, 1/27}]); N[-sm[-1], 105] // RealDigits // First
%t (* or, without using the Weierstrass P function: *) nint[y_?NumericQ] := NIntegrate[1/(1 + w^3)^(2/3), {w, 0, y}, WorkingPrecision -> 105]; smh[t_] := y /. FindRoot[nint[y] == t, {y, t}, WorkingPrecision -> 105]; N[smh[1], 105] // RealDigits // First
%Y Cf. A104133, A104134.
%K nonn,cons,easy
%O 1,2
%A _Jean-François Alcover_, Aug 30 2015
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