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Decimal expansion of a constant related to the asymptotics of A268188 and A333153.
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%I #11 Mar 09 2020 12:10:09

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

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

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

%N Decimal expansion of a constant related to the asymptotics of A268188 and A333153.

%F Equals sqrt(15) * log(phi) / Pi, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

%F If m >= 0 and g.f. is Sum_{k>=1} (k^m * x^(k^2) / Product_{j=1..k} (1 - x^j)), then a(n) ~ A333155^m * phi^(1/2) * exp(2*Pi*sqrt(n/15)) * n^((2*m-3)/4) / (2 * 3^(1/4) * 5^(1/2)).

%F If m >= 0 and g.f. is Sum_{k>=1} (k^m * x^(k*(k+1)) / Product_{j=1..k} (1 - x^j)), then a(n) ~ A333155^m * exp(2*Pi*sqrt(n/15)) * n^((2*m-3)/4) / (2 * 3^(1/4) * 5^(1/2) * phi^(1/2)).

%e 0.5932422150033691271841376173302559541109959549627957429060245786...

%p evalf(sqrt(15) * log((sqrt(5) + 1)/2) / Pi, 120);

%t RealDigits[Sqrt[15]*Log[GoldenRatio]/Pi, 10, 105][[1]]

%Y Cf. A003114, A268188, A333141, A333151, A333152.

%Y Cf. A003106, A333153, A333154.

%K nonn,cons

%O 0,1

%A _Vaclav Kotesovec_, Mar 09 2020