cp's OEIS Frontend

A331502 Decimal expansion of exp(4/9).

%I A331502 #50 Nov 21 2024 09:36:47
%S A331502 1,5,5,9,6,2,3,4,9,7,6,0,6,7,8,0,7,1,5,5,3,3,7,0,9,2,8,6,9,7,9,4,7,1,
%T A331502 1,8,6,3,9,4,8,2,4,0,1,1,4,2,2,1,4,2,3,5,4,3,9,0,2,7,8,4,3,1,5,4,3,5,
%U A331502 6,3,8,5,0,1,3,3,1,0,6,3,2,6,4,2,7,5,8,1,6,1,2,4,9,2,9,9,4,0,1,5,4,2,9,1,6,9
%N A331502 Decimal expansion of exp(4/9).
%C A331502 Considering graph evolutions (see the Flajolet link) with 3n vertices initially isolated, the probability of the occurrence of an acyclic graph at the point n, (n = 1/3 * 3n), in the uniform model, will be denoted by P13(n). In the case of the permutation model, the respective probability will be denoted by Pp13(n).
%C A331502 Pp13(n) / P13(n) ~ exp(4/9) since Pp13(n) = f(n) / C(N,n), where f(n) is the number of labeled forests with 3n nodes and n edges, and C(N,n), N = 3n *(3n-1)/2 (see the Lucatero link) is the number of labeled graphs with 3n nodes and n edges.
%C A331502 Because P13(n) = f(n)* n!* 2^n / (3n)^(2n), Pp13(n) / P13(n) = (3n)^(2n) / (C(N,n)* n! *2^n), and Lim_{n->oo} Pp13(n) / P13(n) = exp(4/9).
%H A331502 P. Flajolet, D. E. Knuth, and B. Pittel, <a href="https://doi.org/10.1016/0012-365X(89)90087-3">The first cycles in an evolving graph</a>, Discrete Mathematics, 75(1-3):167-215, 1989.
%H A331502 Carlos R. Lucatero, <a href="https://www.intechopen.com/online-first/combinatorial-enumeration-of-graphs">Combinatorial Enumeration of Graphs</a>.
%H A331502 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A331502 Equals Lim_{n->oo} Pp13(n) / P13(n) = Lim_{n->oo} (3*n)^(2*n) / (binomial((3*n *(3*n-1)/2), n) * n! * 2^n).
%e A331502 1.55962349760678071553370928697947118639482401142214...
%p A331502 evalf(exp(4/9), 134);
%t A331502 RealDigits[Exp[4/9],10,120][[1]] (* _Harvey P. Dale_, Jun 05 2023 *)
%o A331502 (PARI) exp(4/9)
%Y A331502 Cf. A001113, A019774, A092042, A302112, A331500, A331501, A331505.
%K A331502 nonn,cons
%O A331502 1,2
%A A331502 _Washington Bomfim_, Mar 04 2020