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Triangle read by rows: T(n,k) is the number of unoriented linear forests with n nodes and k rooted trees.
4

%I #11 Dec 08 2020 15:24:31

%S 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,

%T 96,106,69,38,15,4,1,286,239,284,194,124,52,20,4,1,719,622,750,564,

%U 377,189,77,24,5,1,1842,1607,2010,1584,1144,618,292,100,30,5,1

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

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

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

%H Andrew Howroyd, <a href="/A339303/b339303.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)

%F 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.

%e Triangle read by rows:

%e 1;

%e 1, 1;

%e 2, 1, 1;

%e 4, 3, 2, 1;

%e 9, 6, 6, 2, 1;

%e 20, 16, 15, 8, 3, 1;

%e 48, 37, 41, 22, 12, 3, 1;

%e 115, 96, 106, 69, 38, 15, 4, 1;

%e 286, 239, 284, 194, 124, 52, 20, 4, 1;

%e 719, 622, 750, 564, 377, 189, 77, 24, 5, 1;

%e ...

%o (PARI) \\ TreeGf is A000081 as g.f.

%o 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)}

%o 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}

%o M(n, m=n)=Mat(vector(m, k, ColSeq(n,k)~))

%o { my(T=M(12)); for(n=1, #T~, print(T[n,1..n])) }

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

%Y Row sums are A303840(n+2).

%Y Row sums excluding the first column are A303833.

%Y Cf. A339067.

%K nonn,tabl

%O 1,4

%A _Andrew Howroyd_, Dec 04 2020