A332679 -id:A332679 - 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

A277418 a(n) = n!LaguerreL(n, -4n).

Original entry on oeis.org

1, 5, 98, 3246, 151064, 9052120, 663449040, 57490690544, 5749754436992, 651830574374784, 82599621627948800, 11569798584488362240, 1775052172071446510592, 296026752508667034942464, 53320241823337034415908864, 10315767337287172256717568000

Offset: 0

Vaclav Kotesovec, Oct 14 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..250
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials

Cf. A277373, A277391, A277392, A277419, A277420, A277421, A277422.

Cf. A002720, A087912, A277382, A332679.

[Factorial(n)*(&+[Binomial(n,k)*4^k*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, May 15 2018

Table[n!*LaguerreL[n, -4*n], {n, 0, 20}]
Flatten[{1, Table[n!*Sum[Binomial[n, k] * 4^k * n^k / k!, {k, 0, n}], {n, 1, 20}]}]

for(n=0, 30, print1(n!*sum(k=0, n, binomial(n,k)*4^k*n^k/k!), ", ")) \\ G. C. Greubel, May 15 2018

a(n) = n! * Sum_{k=0..n} binomial(n, k) * 4^k * n^k / k!.

a(n) ~ sqrt(2 + 3/sqrt(2)) * (3 + 2*sqrt(2))^n * exp((-3 + 2*sqrt(2))*n) * n^n / 2.

A295408 a(n) = n! * Laguerre(n, 4*n, -n).

Original entry on oeis.org

1, 6, 134, 5052, 267576, 18246850, 1521907056, 150077897088, 17080661438336, 2203559337858174, 317761804144896000, 50650336389453807556, 8843008543955452118016, 1678231571506037926192698, 343989152383931539269349376, 75733086648535784012234565000

Offset: 0

Vaclav Kotesovec, Nov 22 2017

nonn

In general, for fixed m >= 1, n! * Sum_{k=0..n} binomial(m*n, n-k) * n^k / k! = n! * Laguerre(n, (m-1)*n, -n) ~ sqrt(1/2 + (m + 2)/(2*sqrt(m^2 + 4))) * (2^(m+1) * m^m / ((sqrt(m^2 + 4) - m) * (m - 2 + sqrt(m^2 + 4))^m))^n * exp((sqrt(m^2 + 4) - m)*n/2 - n) * n^n.

G. C. Greubel, Table of n, a(n) for n = 0..300
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Cf. A277373 (m=1), A295385 (m=2), A295406 (m=3), A295407 (m=4).

Cf. A295409, A295418, A332679.

[Factorial(n)*(&+[Binomial(5*n,n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 06 2018

Table[n!*LaguerreL[n,4*n,-n],{n,0,15}]
Join[{1},Table[n!*Sum[Binomial[5*n,n-k]*n^k/k!,{k,0,n}],{n,1,15}]]

for(n=0,30, print1(n!*sum(k=0, n, binomial(5*n,n-k)*n^k/k!), ", ")) \\ G. C. Greubel, Feb 06 2018

a(n) = n!*pollaguerre(n, 4*n, -n); \\ Michel Marcus, Feb 05 2021

a(n) = n!*Sum_{k=0..n} binomial(5*n,n-k)*n^k/k!.
a(n) ~ sqrt(1/2 + 7/(2*sqrt(29))) * (131 - 22*sqrt(29))^n * exp((sqrt(29)-7)*n/2) * n^n.
a(n) = n! * [x^n] exp(n*x/(1 - x))/(1 - x)^(4*n+1). - Ilya Gutkovskiy, Nov 23 2017

A332692 a(n) = n! * Laguerre(n, 2*n).

Original entry on oeis.org

1, -1, 2, 6, -232, 4120, -61488, 740432, -3220096, -224705664, 11713068800, -397487915264, 10466018491392, -176186211195904, -2178925657151488, 399827849856768000, -24748326426744881152, 1112888620945558700032, -36293785214959525625856, 408738923015995616067584

Offset: 0

Vaclav Kotesovec, Feb 20 2020

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..350

Cf. A277423, A332693, A332679, A332694, A332695.

Cf. A277391, A295406.

Table[n! * LaguerreL[n, 2*n], {n, 0, 25}]
Flatten[{1, Table[n!*Sum[Binomial[n, k] * (-1)^k * 2^k * n^k / k!, {k, 0, n}], {n, 1, 25}]}]
Table[n! * Hypergeometric1F1[-n, 1, 2*n], {n, 0, 25}]

a(n) = n!*pollaguerre(n, 0, 2*n); \\ Michel Marcus, Feb 05 2021

A332693 a(n) = n! * Laguerre(n, 3*n).

Original entry on oeis.org

1, -2, 14, -156, 2328, -42630, 902736, -20961864, 497925504, -10347816906, 54902188800, 15803663268492, -1741565563831296, 146556727320337074, -11551833579195721728, 901051402625901468000, -71007771313742983888896, 5701873713553516375488366, -467924697090124685492944896

Offset: 0

Vaclav Kotesovec, Feb 20 2020

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..350

Cf. A277423, A332692, A332679, A332694, A332695.

Cf. A277392, A295407.

Table[n! * LaguerreL[n, 3*n], {n, 0, 25}]
Flatten[{1, Table[n!*Sum[Binomial[n, k] * (-1)^k * 3^k * n^k / k!, {k, 0, n}], {n, 1, 25}]}]
Table[n! * Hypergeometric1F1[-n, 1, 3*n], {n, 0, 25}]

a(n) = n!*pollaguerre(n, 0, 3*n); \\ Michel Marcus, Feb 05 2021

A332694 a(n) = (-1)^n * n! * Laguerre(n, 5*n).

Original entry on oeis.org

1, 4, 62, 1614, 58904, 2764880, 158631120, 10755909010, 841471425920, 74605812325020, 7392555309228800, 809594650092540950, 97103822900059929600, 12659189667284189060200, 1782335176686080469555200, 269524635118213823349788250, 43567606796796836119605248000

Offset: 0

Vaclav Kotesovec, Feb 20 2020

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..320

Cf. A277419, A277423, A332692, A332693, A332679, A332695.

Table[(-1)^n * n! * LaguerreL[n, 5*n], {n, 0, 20}]
Flatten[{1, Table[n!*Sum[Binomial[n, k] * (-1)^(n-k) * 5^k * n^k / k!, {k, 0, n}], {n, 1, 20}]}]
Table[(-1)^n * n! * Hypergeometric1F1[-n, 1, 5*n], {n, 0, 20}]

a(n) = (-1)^n*n!*pollaguerre(n, 0, 5*n); \\ Michel Marcus, Feb 05 2021

a(n) ~ exp((3-sqrt(5))*n/2) * ((sqrt(5) + 1)/2)^(2*n+1) * n^n / 5^(1/4). - Vaclav Kotesovec, Feb 20 2020, simplified May 09 2021

A332695 a(n) = (-1)^n * n! * Laguerre(n, 6*n).

Original entry on oeis.org

1, 5, 98, 3234, 149784, 8927880, 650696400, 56061791856, 5574017768832, 628158472212096, 79123082415148800, 11015976349601752320, 1679832851707998600192, 278440504042352431942656, 49846084962712218734045184, 9584526091509128369970432000, 1970059291620925696814892810240

Offset: 0

Vaclav Kotesovec, Feb 20 2020

nonn

For m > 4, (-1)^n * n! * Laguerre(n, m*n) ~ sqrt(1/2 + (m-2)/(2*sqrt(m*(m-4)))) * exp((m - 2 - sqrt(m*(m-4)))*n/2) * ((m - 2 + sqrt(m*(m-4)))/2)^n * n^n.

Vaclav Kotesovec, Table of n, a(n) for n = 0..310

Cf. A277420, A277423, A332692, A332693, A332679, A332694.

Table[(-1)^n * n! * LaguerreL[n, 6*n], {n, 0, 20}]
Flatten[{1, Table[n!*Sum[Binomial[n, k] * (-1)^(n-k) * 6^k * n^k / k!, {k, 0, n}], {n, 1, 20}]}]
Table[(-1)^n * n! * Hypergeometric1F1[-n, 1, 6*n], {n, 0, 20}]

a(n) = (-1)^n*n!*pollaguerre(n, 0, 6*n); \\ Michel Marcus, Feb 05 2021

a(n) ~ sqrt(1/2 + 1/sqrt(3)) * 2^n * exp((2-sqrt(3))*n) * ((1 + sqrt(3))/2)^(2*n) * n^n.

A332680 a(n) = -(-1)^n * n! * hypergeometric1F1(1 - n, 2, 4*n).

Original entry on oeis.org

-1, 1, 6, 78, 1576, 43320, 1507824, 63549808, 3145681536, 178865283456, 11488065875200, 822528662774016, 64957295774721024, 5609010346397166592, 525718830294548330496, 53154054477553828608000, 5766597997397483718344704, 668177890990349738366042112, 82355042760252520538828242944

Offset: 0

Vaclav Kotesovec, Feb 19 2020

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..336

Cf. A302112, A332679.

Table[-(-1)^n * n! * Hypergeometric1F1[1 - n, 2, 4*n], {n, 0, 20}]
Join[{-1}, Table[n! * Sum[(-1)^(n-k+1) * Binomial[n-1, k] * (4*n)^k / (k+1)!, {k, 0, n-1}], {n, 1, 20}]]

A302112(n) = (A332679(n) - 2*n*a(n)) * binomial(2*n, n) / 2^n.

a(n) ~ c * n^(n - 5/6) * exp(n), where c = Gamma(1/3) / (2^(11/6) * 3^(1/6) * sqrt(Pi)) = 0.3531663187295...

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A277418 a(n) = n!*LaguerreL(n, -4*n).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A295408 a(n) = n! * Laguerre(n, 4*n, -n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A332692 a(n) = n! * Laguerre(n, 2*n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A332693 a(n) = n! * Laguerre(n, 3*n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A332694 a(n) = (-1)^n * n! * Laguerre(n, 5*n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A332695 a(n) = (-1)^n * n! * Laguerre(n, 6*n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A332680 a(n) = -(-1)^n * n! * hypergeometric1F1(1 - n, 2, 4*n).

Views

Author

Keywords

Links

A277418 a(n) = n!LaguerreL(n, -4n).