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%I #49 Nov 21 2024 09:31:52
%S 2,1,1,7,0,0,0,0,1,6,6,1,2,6,7,4,6,6,8,5,4,5,3,6,9,8,1,9,8,3,7,0,9,5,
%T 6,1,0,1,3,4,4,9,1,5,8,4,7,0,2,4,0,3,4,2,1,7,7,9,1,3,3,0,3,0,8,1,0,9,
%U 8,4,5,3,3,3,6,4,0,1,2,8,2,0,0,0,2,7,9,1,5,6,0,2,6,6,6,1,5,7,9,8,2,1,8,8,8
%N Decimal expansion of exp(3/4).
%C Considering graph evolutions (see the Flajolet link) with 2n vertices initially isolated, the probability of the occurrence of an acyclic graph at the critical point n in the uniform model, will be denoted by P(n). In the case of the permutation model, the respective probability will be denoted by Pp(n).
%C Pp(n) / P(n) ~ exp(3/4) since Pp(n) = A302112(n) / A331505(2n) = A302112(n) / C(C(2n,2), n), and P(n) = A302112(n) * n! * 2^n / (2n)^(2n), Pp(n) / P(n) = (2n)^(2n) / (C(C(2n,2), n) * n! * 2^n), and lim_{n->oo} Pp(n) / P(n) = exp(3/4).
%H Philippe Flajolet, Donald E. Knuth, and Boris Pittel, <a href="https://doi.org/10.1016/0012-365X(89)90087-3">The first cycles in an evolving graph</a>, Discrete Mathematics, Vol. 75, No. 1-3 (1989), pp. 167-215.
%H Leonard Giugiuc and Dan Stefan Marinescu, <a href="https://cms.math.ca/publications/crux/issue?volume=43&issue=6">Problem 4257</a>, Crux Mathematicorum, Vol. 43, No. 6 (2017), pp. 263 and 265; <a href="https://cms.math.ca/publications/crux/issue?volume=44&issue=6">Solution to Problem 4257</a>, ibid., Vol. 44, No. 6 (2018), pp. 268-270.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Equals lim_{n->oo} Pp(n) / P(n) = lim_{n->oo} (2*n)^(2*n) / (binomial(binomial(2n,2), n) * n! * 2^n).
%F Equals lim_{n->oo} sqrt(n)/A000178(n)^(1/(n*(n+1))) (Giugiuc and Marinescu, 2017). - _Amiram Eldar_, Apr 12 2022
%e 2.1170000166126746685453698198370956101344915847024...
%p evalf(exp(3/4), 134);
%t RealDigits[Exp[3/4], 10, 100][[1]] (* _Amiram Eldar_, Apr 12 2022 *)
%o (PARI) exp(3/4) \\ _Charles R Greathouse IV_, Nov 21 2024
%Y Cf. A000178, A001113, A019774, A092042, A302112, A331500, A331502, A331505.
%K nonn,cons
%O 1,1
%A _Washington Bomfim_, Feb 27 2020