A339067 Triangle read by rows: T(n,k) is the number of linear forests with n nodes and k rooted trees.

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

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.

Also the convolution triangle of A000081. - Peter Luschny, Oct 07 2022

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
# Using function PMatrix from A357368. Adds row and column for n, k = 0.
PMatrix(10, A000081); # Peter Luschny, Oct 07 2022

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