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A249137 Decimal expansion of the derivative y'(0) where y(x) is the solution to the differential equation y''(x)+exp(y(x))=0, with y(0)=y(beta)=0 and beta maximum (beta = A249136).

%I A249137 #16 Feb 16 2025 08:33:24
%S A249137 2,1,3,3,8,7,7,9,3,9,9,1,5,0,6,1,1,1,9,8,0,7,2,4,4,6,7,7,4,0,1,8,5,2,
%T A249137 9,1,9,2,2,8,9,6,2,3,8,5,3,7,9,6,4,6,8,6,1,7,7,7,2,3,4,5,9,2,7,1,9,0,
%U A249137 6,1,1,7,5,5,7,7,4,9,9,0,3,8,1,5,7,5,2,3,9,9,3,3,7,4,7,3,2,9,4,3,3,5,6
%N A249137 Decimal expansion of the derivative y'(0) where y(x) is the solution to the differential equation y''(x)+exp(y(x))=0, with y(0)=y(beta)=0 and beta maximum (beta = A249136).
%H A249137 G. C. Greubel, <a href="/A249137/b249137.txt">Table of n, a(n) for n = 1..10000</a>
%H A249137 Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, p. 32.
%H A249137 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/LaplaceLimit.html">Laplace Limit.</a>
%F A249137 y'(0) = sqrt(2)*sinh(sqrt(lambda^2 + 1)), where lambda is A033259, the Laplace limit constant 0.66274...
%e A249137 2.13387793991506111980724467740185291922896238537964686...
%t A249137 digits = 103; lambda = x /. FindRoot[x*Exp[Sqrt[1 + x^2]]/(1 + Sqrt[1 + x^2]) == 1, {x, 1}, WorkingPrecision -> digits+5]; mu = Sqrt[lambda^2 + 1]; RealDigits[Sqrt[2]*Sinh[mu], 10, digits] // First
%Y A249137 Cf. A033259, A085984, A248916, A249136.
%K A249137 nonn,cons,easy
%O A249137 1,1
%A A249137 _Jean-François Alcover_, Oct 22 2014