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 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). 3
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The first formula is based on Kolchin's formula (1.4.2) [see the Kolchin reference]. 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) 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). EXAMPLE 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. PROG (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 CROSSREFS Columns k=2..3 are A001147, A332960. Row sums give A302112. Main diagonal is A332958. Cf. A000272, A001147, A302112, A332958, A332960. Sequence in context: A361925 A096854 A013191 * A009783 A341013 A192786 Adjacent sequences: A332956 A332957 A332958 * A332960 A332961 A332962 KEYWORD nonn,tabl,easy AUTHOR Washington Bomfim, Apr 13 2020 STATUS approved

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Last modified June 22 15:32 EDT 2024. Contains 373587 sequences. (Running on oeis4.)