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 A144289 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows: Number T(n,k) of forests of labeled rooted trees on n or fewer nodes using a subset of labels 1..n and k edges. 3
 1, 2, 0, 4, 2, 0, 8, 12, 9, 0, 16, 48, 84, 64, 0, 32, 160, 480, 820, 625, 0, 64, 480, 2160, 6120, 10230, 7776, 0, 128, 1344, 8400, 34720, 94500, 155274, 117649, 0, 256, 3584, 29568, 165760, 647920, 1712592, 2776200, 2097152, 0, 512, 9216, 96768, 701568, 3669120, 13783392, 35630784, 57138120, 43046721, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Alois P. Heinz, Rows n = 0..140, flattened FORMULA T(n,0) = 2^n, T(n,k) = 0 if k < 0 or n <= k, otherwise T(n,k) = n^(n-1) if k=n-1, otherwise T(n,k) = Sum_{j=0..k} C(n-1,j)*T(j+1,j)*T(n-1-j,k-j). EXAMPLE T(3,1) = 12, because there are 12 forests of labeled rooted trees on 3 or fewer nodes using a subset of labels 1..3 and 1 edge:   .1<2. .2<1. .1<3. .3<1. .2<3. .3<2. .1<2. .2<1. .1<3. .3<1. .2<3. .3<2.   ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... .....   ..... ..... ..... ..... ..... ..... .3... .3... .2... .2... .1... .1... Triangle begins:    1;    2,   0;    4,   2,   0;    8,  12,   9,   0;   16,  48,  84,  64,   0;   32, 160, 480, 820, 625,  0; MAPLE T:= proc(n, k) option remember;       if k=0 then 2^n     elif k<0 or n<=k then 0     elif k=n-1 then n^(n-1)     else add(binomial(n-1, j) *T(j+1, j) *T(n-1-j, k-j), j=0..k)       fi     end: seq(seq(T(n, k), k=0..n), n=0..11); MATHEMATICA T[n_, k_] := T[n, k] = Which[k == 0, 2^n, k<0 || n <= k, 0, k == n-1, n^(n-1), True, Sum[Binomial[n-1, j]*T[j+1, j]*T[n-1-j, k-j], {j, 0, k}]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 11}] // Flatten (* Jean-François Alcover, Jan 21 2014, translated from Alois P. Heinz's Maple code *) CROSSREFS Columns 0, 1 give A000079, A001815. First lower diagonal gives A000169 with first term 2. Row sums give A088957. Cf. A007318, A000142. Sequence in context: A341419 A185879 A081880 * A293815 A339941 A211318 Adjacent sequences:  A144286 A144287 A144288 * A144290 A144291 A144292 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 17 2008 STATUS approved

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Last modified June 15 15:16 EDT 2021. Contains 345049 sequences. (Running on oeis4.)