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A143398 Triangle T(n,k) = number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains k labels, n>=0, 0<=k<=n.

%I A143398 #31 May 13 2022 18:20:52
%S A143398 1,0,1,0,3,1,0,10,3,1,0,41,9,4,1,0,196,40,10,5,1,0,1057,210,30,15,6,1,
%T A143398 0,6322,1176,175,35,21,7,1,0,41393,7273,1176,105,56,28,8,1,0,293608,
%U A143398 49932,7084,756,126,84,36,9,1,0,2237921,372060,42120,6510,378,210,120,45,10,1
%N A143398 Triangle T(n,k) = number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains k labels, n>=0, 0<=k<=n.
%H A143398 Alois P. Heinz, <a href="/A143398/b143398.txt">Rows n = 0..140, flattened</a>
%H A143398 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A143398 T(n,k) = n! * Sum_{i=0..u(n,k)} i^(n-k*i)/((n-k*i)!*i!*k!^i) with u(n,k) = 0 if k=0 and u(n,k) = floor(n/k) else.
%e A143398 T(4,2) = 9: 3->{1,2}<-4, 2->{1,3}<-4, 2->{1,4}<-3, 1->{2,3}<-4, 1->{2,4}<-3, 1->{3,4}<-2, {1,2}{3,4}, {1,3}{2,4}, {1,4}{2,3}.
%e A143398 Triangle begins:
%e A143398   1;
%e A143398   0,   1;
%e A143398   0,   3,  1;
%e A143398   0,  10,  3,  1;
%e A143398   0,  41,  9,  4,  1;
%e A143398   0, 196, 40, 10,  5,  1;
%e A143398   ...
%p A143398 u:= (n, k)-> `if`(k=0, 0, floor(n/k)):
%p A143398 T:= (n, k)-> n! *add(i^(n-k*i)/ ((n-k*i)! *i! *k!^i), i=0..u(n, k)):
%p A143398 seq(seq(T(n, k), k=0..n), n=0..12);
%t A143398 t[n_, n_] = 1; t[_, 0] = 0; t[n_, k_] := n!*Sum[i^(n-k*i)/((n-k*i)!*i!*k!^i), {i, 0, n/k}]; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 19 2013 *)
%o A143398 (PARI) u(n,k) = if(k==0, 0, n\k);
%o A143398 T(n, k) = n!*sum(j=0, u(n, k), j^(n-k*j)/(k!^j*j!*(n-k*j)!)); \\ _Seiichi Manyama_, May 13 2022
%Y A143398 Columns k=0-9 give: A000007, A000248, A133189, A145453, A145454, A145455, A145456, A145457, A145458, A145459.
%Y A143398 Main diagonal gives A000012.
%Y A143398 Row sums give A143406.
%Y A143398 T(2n,n) gives A029651.
%Y A143398 Cf. A000142, A145460, A351703.
%K A143398 nonn,tabl,look
%O A143398 0,5
%A A143398 _Alois P. Heinz_, Aug 12 2008