A332959 Triangle read by rows: T(n,k) is the number of labeled forests with n trees and 2n nodes and with the largest tree having exactly k nodes, (n >= 1, 2 <= k <= n+1).
1, 3, 12, 15, 180, 240, 105, 5040, 6720, 7000, 945, 151200, 352800, 315000, 272160, 10395, 6029100, 21067200, 20790000, 17962560, 13311144, 135135, 276215940, 1387386000, 1765764000, 1471133664, 1211314104, 787218432, 2027025, 14983768800, 105945840000, 165225060000, 146023637760, 121131410400, 94466211840, 54717165360
Offset: 1
Examples
Triangle T(n,k) begins:
1;
3, 12;
15, 180, 240;
105, 5040, 6720, 7000;
945, 151200, 352800, 315000, 272160;
10395, 6029100, 21067200, 20790000, 17962560, 13311144;
...
The graphs for T(2,2) and T(2,3) are illustrated below:
o---o : o o
: |
o---o : o---o
T(2,2) = 3 since the graph on the left has 3 labelings.
T(2,3) = 12 since the graph on the right has 12 labelings.
References
- V. F. Kolchin, Random Graphs. Encyclopedia of Mathematics and Its Applications 53. Cambridge Univ. Press, Cambridge, 1999, pp 30-31.
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
Crossrefs
Programs
-
PARI
T(n, k) = { my(S = 0); forpart(a = 2*n, if(a[n] == k, my(D = Set(a)); my(Pr = prod(j=1, #D, my(p = D[j], c = #select(x->x==p, Vec(a))); p^((p-2)*c) / (p!^c*c!))); S += n!*Pr ) , [1, k], [n, n]); (2*n)! / n! * S }; -
PARI
B(n,k)={my(p=sum(j=1, k, j^(j-2)*x^j/j!)); (2*n)!*polcoef( polcoef( exp(y*p + O(x*x^(2*n))), 2*n, x), n, y)} T(n,k)={B(n,k)-B(n,k-1)} \\ Andrew Howroyd, May 08 2020
Formula
T(n,k) = ((2*n)!/n!) * Sum_{compositions p_1 + ... + p_n = 2*n, 1 <= p_i <= k}
Product_{j=1..n} f(p_j) / p_j!, where f(p_j) = A000272(p_j) = p_j^(p_j-2).
Comments