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A245655 Decimal expansion of eta_A, a constant associated with the asymptotics of the enumeration of labeled acyclic digraphs. 0

%I #15 Jan 21 2015 17:33:00

%S 5,7,4,3,6,2,3,7,3,3,0,9,3,1,1,4,7,6,9,1,6,6,7,0,8,0,1,6,8,1,5,0,7,2,

%T 4,6,9,7,2,1,8,8,4,6,0,9,7,0,8,7,5,4,2,4,0,6,9,0,2,2,4,7,9,1,2,2,0,2,

%U 8,6,8,9,4,0,3,7,1,7,7,3,3,7,7,1,5,7,3,8,0,5,2,5,3,5,4,1,0,6,4,2,6,9,4

%N Decimal expansion of eta_A, a constant associated with the asymptotics of the enumeration of labeled acyclic digraphs.

%C The article "Acyclic digraphs and eigenvalues of (0,1)-matrices" gives the wrong value M=0.474! See A003024 for more. - _Vaclav Kotesovec_, Jul 28 2014

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.6 Otter's Tree Enumeration Constants, p. 310.

%H B. D. McKay, F. E. Oggier, G. F. Royle, N. J. A. Sloane, I. M. Wanless and H. S. Wilf, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL7/Sloane/sloane15.html">Acyclic digraphs and eigenvalues of (0,1)-matrices</a>, J. Integer Sequences, 7 (2004), #04.3.3.

%H B. D. McKay, F. E. Oggier, G. F. Royle, N. J. A. Sloane, I. M. Wanless and H. S. Wilf, <a href="http://arXiv.org/abs/math.CO/0310423">Acyclic digraphs and eigenvalues of (0,1)-matrices</a>, arXiv:math.CO/0310423 (2003)

%F eta_A = xi_A*lambda(xi_A/2), where xi_A is the smallest positive root of lambda(x) = sum_{n >= 0} (-1)^n*x^n/(2^(n*(n-1)/2)*n!).

%e 0.5743623733093114769166708016815072469721884609708754240690224791220286894...

%t digits = 103; lambda[x_?NumericQ] := NSum[(-1)^n*x^n/(2^(n*(n - 1)/2)*n!), {n, 0, Infinity}, WorkingPrecision -> digits + 10, Method -> "AlternatingSigns"]; xi = x /. FindRoot[lambda[x] == 0, {x, 3/2}, WorkingPrecision -> digits + 10]; RealDigits[xi*lambda[xi/2], 10, digits] // First

%Y Cf. A245654, A003024.

%K nonn,cons

%O 0,1

%A _Jean-François Alcover_, Jul 28 2014

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