A339303 Triangle read by rows: T(n,k) is the number of unoriented linear forests with n nodes and k rooted trees.

1, 1, 1, 2, 1, 1, 4, 3, 2, 1, 9, 6, 6, 2, 1, 20, 16, 15, 8, 3, 1, 48, 37, 41, 22, 12, 3, 1, 115, 96, 106, 69, 38, 15, 4, 1, 286, 239, 284, 194, 124, 52, 20, 4, 1, 719, 622, 750, 564, 377, 189, 77, 24, 5, 1, 1842, 1607, 2010, 1584, 1144, 618, 292, 100, 30, 5, 1

Offset: 1

Author

Andrew Howroyd, Dec 04 2020

Keywords

• nonn
• tabl

Comments

Linear forests (A339067) are considered up to reversal of the linear order.

T(n,k) is the number of unlabeled trees on n nodes rooted at two indistinguishable nodes at distance k-1 from each other.

Examples

			Triangle read by rows:
    1;
    1,   1;
    2,   1,   1;
    4,   3,   2,   1;
    9,   6,   6,   2,   1;
   20,  16,  15,   8,   3,   1;
   48,  37,  41,  22,  12,   3,  1;
  115,  96, 106,  69,  38,  15,  4,  1;
  286, 239, 284, 194, 124,  52, 20,  4, 1;
  719, 622, 750, 564, 377, 189, 77, 24, 5, 1;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Crossrefs

Columns 1..4 are A000081, A027852, A280788(n-3), A339302.

Row sums are A303840(n+2).

Row sums excluding the first column are A303833.

Cf. A339067.

Programs

PARI

\\ TreeGf is A000081 as g.f.
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(r=TreeGf(max(0,n+1-k))); Vec(r^k + r^(k%2)*subst(r, x, x^2)^(k\2), -n)/2}
M(n, m=n)=Mat(vector(m, k, ColSeq(n,k)~))
{ my(T=M(12)); for(n=1, #T~, print(T[n,1..n])) }

Formula

G.f of column k: (r(x)^k + r(x)^(k mod 2)*r(x^2)^floor(k/2))/2 where r(x) is the g.f. of A000081.