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 A339067 Triangle read by rows: T(n,k) is the number of linear forests with n nodes and k rooted trees. 11
 1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 9, 12, 9, 4, 1, 20, 30, 25, 14, 5, 1, 48, 74, 69, 44, 20, 6, 1, 115, 188, 186, 133, 70, 27, 7, 1, 286, 478, 503, 388, 230, 104, 35, 8, 1, 719, 1235, 1353, 1116, 721, 369, 147, 44, 9, 1, 1842, 3214, 3651, 3168, 2200, 1236, 560, 200, 54, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS T(n,k) is the number of trees with n nodes rooted at two noninterchangeable nodes at a distance k-1 from each other. LINKS Alois P. Heinz, Rows n = 1..200, flattened FORMULA G.f. of k-th column: t(x)^k where t(x) is the g.f. of A000081. Sum_{k=1..n} k * T(n,k) = A038002(n). - Alois P. Heinz, Dec 04 2020 EXAMPLE Triangle begins:     1;     1,    1;     2,    2,    1;     4,    5,    3,    1;     9,   12,    9,    4,   1;    20,   30,   25,   14,   5,   1;    48,   74,   69,   44,  20,   6,   1;   115,  188,  186,  133,  70,  27,   7,  1;   286,  478,  503,  388, 230, 104,  35,  8, 1;   719, 1235, 1353, 1116, 721, 369, 147, 44, 9, 1;   ... MAPLE b:= proc(n) option remember; `if`(n<2, n, (add(add(d*b(d),       d=numtheory[divisors](j))*b(n-j), j=1..n-1))/(n-1))     end: T:= proc(n, k) option remember; `if`(k=1, b(n), (t->       add(T(j, t)*T(n-j, k-t), j=1..n-1))(iquo(k, 2)))     end: seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Dec 04 2020 MATHEMATICA b[n_] := b[n] = If[n < 2, n, (Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n - j], {j, 1, n - 1}])/(n - 1)]; T[n_, k_] := T[n, k] = If[k == 1, b[n], With[{t = Quotient[k, 2]}, Sum[T[j, t]*T[n - j, k - t], {j, 1, n - 1}]]]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jan 03 2021, after Alois P. Heinz *) PROG (PARI) \\ TreeGf is A000081. TreeGf(N) = {my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)} ColSeq(n, k)={my(t=TreeGf(max(0, n+1-k))); Vec(t^k, -n)} M(n, m=n)=Mat(vector(m, k, ColSeq(n, k)~)) { my(T=M(12)); for(n=1, #T~, print(T[n, 1..n])) } CROSSREFS Columns 1..6 are A000081, A000106, A000242, A000300, A000343, A000395. Row sums are A000107. T(2n-1,n) gives A339440. Cf. A033185, A038002, A217781, A339428. Sequence in context: A105306 A183191 A273713 * A322329 A064189 A273897 Adjacent sequences:  A339064 A339065 A339066 * A339068 A339069 A339070 KEYWORD nonn,tabl AUTHOR Andrew Howroyd, Dec 03 2020 STATUS approved

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Last modified July 24 05:05 EDT 2021. Contains 346273 sequences. (Running on oeis4.)