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A143396 Triangle T(n,k) = number of forests of labeled rooted trees of height at most 1, with n labels, k of which are used for root nodes and any root may contain >= 1 labels, n >= 0, 0<=k<=n.

%I A143396 #39 Jan 14 2021 17:35:32
%S A143396 1,0,1,0,2,2,0,3,9,5,0,4,30,40,15,0,5,90,220,185,52,0,6,255,1040,1485,
%T A143396 906,203,0,7,693,4550,9905,9891,4718,877,0,8,1820,19040,59850,87416,
%U A143396 66808,26104,4140,0,9,4644,77448,341082,686826,750120,463212,153063,21147
%N A143396 Triangle T(n,k) = number of forests of labeled rooted trees of height at most 1, with n labels, k of which are used for root nodes and any root may contain >= 1 labels, n >= 0, 0<=k<=n.
%H A143396 Alois P. Heinz, <a href="/A143396/b143396.txt">Rows n = 0..140, flattened</a>
%H A143396 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A143396 T(n,k) = C(n,k) * Sum_{t=0..k} Stirling2(k,t) * t^(n-k).
%F A143396 E.g.f.: exp(exp(x)*(exp(x*y)-1)). - _Vladeta Jovovic_, Dec 08 2008
%e A143396 T(3,2) = 9: {1,2}<-3, {1,3}<-2, {2,3}<-1, {1}<-3{2}, {1}{2}<-3, {1}<-2{3}, {1}{3}<-2, {2}<-1{3}, {2}{3}<-1.
%e A143396 Triangle begins:
%e A143396   1;
%e A143396   0,  1;
%e A143396   0,  2,  2;
%e A143396   0,  3,  9,   5;
%e A143396   0,  4, 30,  40,  15;
%e A143396   0,  5, 90, 220, 185,  52;
%e A143396   ...
%p A143396 T:= (n, k)-> binomial(n, k)*add(Stirling2(k, t)*t^(n-k), t=0..k):
%p A143396 seq(seq(T(n, k), k=0..n), n=0..11);
%t A143396 T[n_, k_] := T[n, k] = Binomial[n, k]*Sum[StirlingS2[k, t]*If[n == k, 1, t^(n - k)], {t, 0, k}];
%t A143396 Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 27 2016, translated from Maple, updated Jan 01 2021 *)
%Y A143396 Columns k=0-10 give: A000007, A000027, A273652, A273653, A273654, A273655, A273656, A273657, A273658, A273659, A273660.
%Y A143396 Diagonal gives A000110.
%Y A143396 Row sums give A143405.
%Y A143396 T(2n,n) gives A273661.
%Y A143396 Cf. A048993, A008277, A007318.
%K A143396 nonn,tabl
%O A143396 0,5
%A A143396 _Alois P. Heinz_, Aug 12 2008