A331500 -id:A331500 - cp's OEIS frontend

A302112 Number of forests with 2n nodes and n labeled trees. Also number of forests with exactly n edges on 2n labeled nodes.

1, 1, 15, 435, 18865, 1092105, 79170399, 6899167275, 702495121185, 81857181636945, 10742799174110575, 1568060617808784099, 251983549987815976785, 44207398164005846558425, 8407483858740005340602175, 1722961754698440157865926875, 378507890849998531093971032385

Offset: 0

Alois P. Heinz, Apr 01 2018

nonn

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)

Alois P. Heinz, Table of n, a(n) for n = 0..310
P. Flajolet, D. E. Knuth, and B. Pittel, The first cycles in an evolving graph, Discrete Mathematics, 75(1-3):167-215, 1989.

Cf. A000272, A138464, A331500, A331505.

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);

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 *)

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

A331505 Number of labeled graphs with n nodes and floor(n/2) edges.

Original entry on oeis.org

1, 1, 3, 15, 45, 455, 1330, 20475, 58905, 1221759, 3478761, 90858768, 256851595, 8093990190, 22760723700, 840261910995, 2353351951665, 99615373765775, 278110855548955, 13278694407181203, 36976937738226486

Offset: 1

Washington Bomfim, Jan 18 2020

Considering the permutation model of graph evolution (see the Flajolet reference) with 2n vertices initially isolated, the probability of the occurrence of an acyclic graph at the critical point n is Pp(n) = A302112(n)/a(2n). Note that a(2n) is the number of labeled graphs with 2n nodes and n edges.
Since a(2n) = C(C(2n, 2), n) we have Pp(n)= A302112(n)/C(C(2n, 2), n).
Therefore, by Vaclav Kotesovec's approximation in A302112, Pp(n) ~ e^(3/4) * P(n), where P(n) = c1 / n^(1/6) is the corresponding probability in the uniform model. Cf. A331500.
If t < n, P(n) is a lower bound of P(t). If t > n, P(n) is an upper bound of P(t). Here P(t) is the probability of an acyclic graph in time t.
Concerning the permutation model, the presence of cycles in graphs evolving near the critical time should be estimated by the above approximation.

			a(4) is 15 because for n = 4, floor(n/2) = 2, and there are two graphs with four points and two edges. See the figure below or the J. Riordan reference.
The non-isomorphic graphs with four nodes and two edges along with the corresponding number of labeled graphs are as follows:
.
  *--*     *  *
  |        |  |
  |        |  |
  *  *     *  *
   12        3
Pp(2) = A302112(2)/a(4) = 15/15 = 1. All the graphs with four nodes and two edges are acyclic.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 109.

Washington Bomfim, Approximation of Pp(n)
P. Flajolet, D. E. Knuth, and B. Pittel, The first cycles in an evolving graph, Discrete Mathematics, 75(1-3):167-215, 1989.
Carlos R. Lucatero, Combinatorial Enumeration of Graphs.

Cf. A000717, A084546, A331504, A302112 (numerators of Pp(n)), A331500, A331501.

C(x, y) = binomial(x, y);
a(n) = C(C(n,2), n\2);
A302112(n)={my(S=0, j); /* From Jon E. Schoenfield's formula in A302112. */
for(j = 0, n,
   S+=(-1/2)^j* C(n, j) * C(2*n-1, n+j-1) * (2*n)^(n-j) * (n+j)!
   );
(1/n!)*S
}; /* end A302112(n) */
c1 = (2/3)^(1/3) * sqrt(Pi) / gamma(1/3);
UpBoundP(n) = c1 / n^(1/6);            /* Approximation for P(n) */
UpBoundPp(n) = exp(3/4) * UpBoundP(n); /* Approximation for Pp(n) */
Pp(n) = A302112(n)/a(2*n);
Ratio(n) = UpBoundPp(n) / Pp(n);

a(n) = C( C(n,2), floor(n/2) ).

A331501 Decimal expansion of exp(3/4).

Original entry on oeis.org

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, 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, 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

Offset: 1

Washington Bomfim, Feb 27 2020

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).

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).

			2.1170000166126746685453698198370956101344915847024...

Philippe Flajolet, Donald E. Knuth, and Boris Pittel, The first cycles in an evolving graph, Discrete Mathematics, Vol. 75, No. 1-3 (1989), pp. 167-215.
Leonard Giugiuc and Dan Stefan Marinescu, Problem 4257, Crux Mathematicorum, Vol. 43, No. 6 (2017), pp. 263 and 265; Solution to Problem 4257, ibid., Vol. 44, No. 6 (2018), pp. 268-270.
Index entries for transcendental numbers

Cf. A000178, A001113, A019774, A092042, A302112, A331500, A331502, A331505.

Maple
```
evalf(exp(3/4), 134);
```

RealDigits[Exp[3/4], 10, 100][[1]] (* Amiram Eldar, Apr 12 2022 *)

exp(3/4) \\ Charles R Greathouse IV, Nov 21 2024

Equals lim_{n->oo} Pp(n) / P(n) = lim_{n->oo} (2*n)^(2*n) / (binomial(binomial(2n,2), n) * n! * 2^n).

Equals lim_{n->oo} sqrt(n)/A000178(n)^(1/(n*(n+1))) (Giugiuc and Marinescu, 2017). - Amiram Eldar, Apr 12 2022

A331502 Decimal expansion of exp(4/9).

Original entry on oeis.org

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, 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, 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

Offset: 1

Washington Bomfim, Mar 04 2020

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).
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.
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).

			1.55962349760678071553370928697947118639482401142214...

P. Flajolet, D. E. Knuth, and B. Pittel, The first cycles in an evolving graph, Discrete Mathematics, 75(1-3):167-215, 1989.
Carlos R. Lucatero, Combinatorial Enumeration of Graphs.
Index entries for transcendental numbers

Cf. A001113, A019774, A092042, A302112, A331500, A331501, A331505.

Maple
```
evalf(exp(4/9), 134);
```

RealDigits[Exp[4/9],10,120][[1]] (* Harvey P. Dale, Jun 05 2023 *)

PARI
```
exp(4/9)
```

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).

cp's OEIS Frontend

A302112 Number of forests with 2n nodes and n labeled trees. Also number of forests with exactly n edges on 2n labeled nodes.

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A331505 Number of labeled graphs with n nodes and floor(n/2) edges.

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A331501 Decimal expansion of exp(3/4).

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A331502 Decimal expansion of exp(4/9).

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