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A225465 Triangular array read by rows: T(n, k) is the number of rooted forests on {1, 2, ..., n} in which one tree has been specially designated that contain exactly k trees; n >= 1, 1 <= k <= n.

%I A225465 #21 Feb 22 2025 06:39:46
%S A225465 1,2,2,9,12,3,64,96,36,4,625,1000,450,80,5,7776,12960,6480,1440,150,6,
%T A225465 117649,201684,108045,27440,3675,252,7,2097152,3670016,2064384,573440,
%U A225465 89600,8064,392,8,43046721,76527504,44641044,13226976,2296350,244944,15876,576,9
%N A225465 Triangular array read by rows: T(n, k) is the number of rooted forests on {1, 2, ..., n} in which one tree has been specially designated that contain exactly k trees; n >= 1, 1 <= k <= n.
%C A225465 Row sums = 2n*(n+1)^(n-2) = A089946(offset).
%C A225465 The average number of trees in each forest approaches 5/2 as n gets large.
%C A225465 The rows give the coefficients of the derivatives of the Abel polynomials. - _Peter Luschny_, Feb 22 2025
%F A225465 T(n, k) = binomial(n-1, k-1)*n^(n-k)*k = A061356(n, k)*k(offset).
%F A225465 E.g.f.: y*A(x)*exp(y*A(x)) where A(x) is e.g.f. for A000169.
%e A225465     T(2,1)=2                  T(2,2)=2
%e A225465   ...1'...   ...2'...   ...1'..2...   ...1..2'...
%e A225465   ...| ...   ...| ...   ...........   ...........
%e A225465   ...2 ...   ...1 ...   ...........   ...........
%e A225465 The root node is on top.  The ' indicates the tree which has been specially designated.
%e A225465 Triangle starts:
%e A225465   [1]        1;
%e A225465   [2]        2,        2;
%e A225465   [3]        9,       12,        3;
%e A225465   [4]       64,       96,       36,        4;
%e A225465   [5]      625,     1000,      450,       80,       5;
%e A225465   [6]     7776,    12960,     6480,     1440,     150,      6;
%e A225465   [7]   117649,   201684,   108045,    27440,    3675,    252,     7;
%e A225465   [8]  2097152,  3670016,  2064384,   573440,   89600,   8064,   392,   8;
%e A225465   [9] 43046721, 76527504, 44641044, 13226976, 2296350, 244944, 15876, 576, 9;
%t A225465 Table[Table[Binomial[n - 1, k - 1] n^(n - k) k, {k, 1, n}], {n, 1, 8}] // Grid
%Y A225465 Cf. A061356, A089946 (row sums), A000169, A137452.
%K A225465 nonn,tabl
%O A225465 1,2
%A A225465 _Geoffrey Critzer_, May 08 2013