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If k > floor(n/2), T(n,k) = 0; otherwise T(n,k) = a(n-k, k), a(n,k) the triangle given by A033185.

E. M. Palmer and A. J. Schwenk, <a href="httphttps://dx.doi.org/10.1016/0095-8956(79)90073-X">On the number of trees in a random forest</a>, J. Combin. Theory, B 27 (1979), 109-121.

<a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

If k > floor(n/2), T(n,k) = 0; otherwise T(n,k) = A033185(n-k, k).

Cf. A000081, A005199, A005198 (row sums), A033185.

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The number of planted trees with n+1 nodes is equal to the number of rooted trees with n nodes. [See Palmer-Schwenk link , pp . 115].

If k > floor(n/2) , T(n,k) = 0 ; otherwise T(n,k) = a(n-k, k), a(n,k) the triangle given by A033185.

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n\t k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

n\t k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

FTriangle read by rows: T(n,t k) is the number of forests with n (unlabeled) nodes and exactly t k planted trees.

If t k > floor(n/2) FT(n,tk) = 0 otherwise FT(n,tk) = a(n-t, tk, k), a(n,tk) the triangle given by A033185.

FT(1,1) = 0, if n >= 2 FT(n,tk) = Sum_{P_1(n,tk)}( Product_{kj=2..n} binomial(A000081(kj-1) + c_k j - 1, c_kj) ), where P_1(n, tk) is the set of the partitions of n with t k parts greater than one: 2*c_2 + ... + n*c_n = n; c_2, ..., c_n >= 0.

Triangle FT(n,tk)

15 32973, 11185, 2095, 320, 47, 7, 1, 0, 0, 0, 0, 0, 0, 0, 0...;

...

A005199(6) = Sum_t t{k=1..6}( k *F T(6,tk) ) = 1*9 + 2*3 +3*1 = 18.

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6 9, 3, 1, 0, 0, 0; A005199(6)=Sum_t t*F(6,t)= 1*9 + 2*3 +3*1 = 18.

6 9, 3, 1, 0, 0, 0;

A005199(6) = Sum_t t*F(6,t) = 1*9 + 2*3 +3*1 = 18.