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A302112 Number of forests with 2n nodes and n labeled trees. Also number of forests with exactly n edges on 2n labeled nodes. 14
1, 1, 15, 435, 18865, 1092105, 79170399, 6899167275, 702495121185, 81857181636945, 10742799174110575, 1568060617808784099, 251983549987815976785, 44207398164005846558425, 8407483858740005340602175, 1722961754698440157865926875, 378507890849998531093971032385 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
From Washington Bomfim, Mar 20 2020: (Start)
Considering the uniform model of graph evolution [see the Flajolet link] with 2n vertices initially isolated, the probability of the occurrence of an acyclic graph at the critical point n is P(n) = a(n) * n! * 2^n / (2n)^(2n). Concerning the permutation model [see same link] the corresponding probability is Pp(n) = a(n) / A331505(2n).
By Kotesovec's approximation of a(n), P(n) ~ c1/n^(1/6), and Pp(n) ~ e^(3/4)* P(n), c1 = 0.577983047665... = (2/3)^(1/3) * sqrt(Pi) / Gamma(1/3).
In both models the presence of cycles in graphs evolving near the critical time should be estimated by the above approximations. (End)
LINKS
P. Flajolet, D. E. Knuth, and B. Pittel, The first cycles in an evolving graph, Discrete Mathematics, 75(1-3):167-215, 1989.
FORMULA
a(n) = A105599(2*n,n) = A138464(2*n,n).
a(n) ~ c * 2^n * exp(n) * n^(n - 2/3), where c = 0.2305818... = 1 / (2^(1/6) * 3^(1/3) * Gamma(1/3)) [symbolic expression for c is conjectural]. - Vaclav Kotesovec, Jul 20 2019, updated Feb 20 2020
a(n) = (1/n!) * Sum_{j=0..n} (-1/2)^j * binomial(n,j) * binomial(2*n-1,n+j-1) * (2*n)^(n-j) * (n+j)!. - Jon E. Schoenfield, Jan 13 2020
a(n) = (-1)^n * (2*n)! * (Laguerre(n, 4*n) + 2*n*hypergeometric1F1(1 - n, 2, 4*n)) / (n! * 2^n). - Vaclav Kotesovec, Feb 19 2020
a(n) = (A332679(n) - 2*n*A332680(n)) * binomial(2*n, n) / 2^n. - Vaclav Kotesovec, Feb 20 2020
a(n) = (2*n)! * Sum_{P(2*n,n)} Product_{p=1..2*n} f(p)^c_p / (c_p! * p!^c_p), where f(n) = A000272(n) = n^(n-2) and P(2*n,n) are the partitions of 2*n with n parts, 1*c_1 + 2*c_2 + ... + (2*n)*c_n; c_1, c_2, ..., c_(2*n) >= 0.
- Washington Bomfim, Apr 05 2020
MAPLE
T:= proc(n, m) option remember; `if`(n<0, 0, `if`(n=m, 1,
`if`(m<1 or m>n, 0, add(binomial(n-1, j-1)*j^(j-2)*
T(n-j, m-1), j=1..n-m+1))))
end:
a:= n-> T(2*n, n):
seq(a(n), n=0..20);
MATHEMATICA
Flatten[{1, Table[Sum[(-1)^k * Binomial[n, k] * Binomial[2*n - 1, n - k] * 2^(n - 2*k) * n^(n - k) * (n + k)!, {k, 0, n} ] / n!, {n, 1, 20}]}] (* Vaclav Kotesovec, Jul 19 2019 *)
Table[(-1)^n * HypergeometricPFQ[{1 - 2*n, -n}, {1, -2*n}, 4*n] * (2*n)! / (n!*2^n), {n, 0, 20}] (* Vaclav Kotesovec, Jul 19 2019 *)
Table[(-1)^n * 2^n * Gamma[n + 1/2] * (2*n*Hypergeometric1F1[1 - n, 2, 4*n] + LaguerreL[n, 4*n]) / Sqrt[Pi], {n, 0, 20}] (* Vaclav Kotesovec, Feb 19 2020 *)
CROSSREFS
Sequence in context: A361284 A323781 A253447 * A262077 A225492 A256194
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Apr 01 2018
STATUS
approved

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Last modified June 15 10:19 EDT 2024. Contains 373407 sequences. (Running on oeis4.)