cp's OEIS Frontend

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

%I A339303 #11 Dec 08 2020 15:24:31
%S A339303 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 A339303 96,106,69,38,15,4,1,286,239,284,194,124,52,20,4,1,719,622,750,564,
%U A339303 377,189,77,24,5,1,1842,1607,2010,1584,1144,618,292,100,30,5,1
%N A339303 Triangle read by rows: T(n,k) is the number of unoriented linear forests with n nodes and k rooted trees.
%C A339303 Linear forests (A339067) are considered up to reversal of the linear order.
%C A339303 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 A339303 Andrew Howroyd, <a href="/A339303/b339303.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%F A339303 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 A339303 Triangle read by rows:
%e A339303     1;
%e A339303     1,   1;
%e A339303     2,   1,   1;
%e A339303     4,   3,   2,   1;
%e A339303     9,   6,   6,   2,   1;
%e A339303    20,  16,  15,   8,   3,   1;
%e A339303    48,  37,  41,  22,  12,   3,  1;
%e A339303   115,  96, 106,  69,  38,  15,  4,  1;
%e A339303   286, 239, 284, 194, 124,  52, 20,  4, 1;
%e A339303   719, 622, 750, 564, 377, 189, 77, 24, 5, 1;
%e A339303   ...
%o A339303 (PARI) \\ TreeGf is A000081 as g.f.
%o A339303 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 A339303 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 A339303 M(n, m=n)=Mat(vector(m, k, ColSeq(n,k)~))
%o A339303 { my(T=M(12)); for(n=1, #T~, print(T[n,1..n])) }
%Y A339303 Columns 1..4 are A000081, A027852, A280788(n-3), A339302.
%Y A339303 Row sums are A303840(n+2).
%Y A339303 Row sums excluding the first column are A303833.
%Y A339303 Cf. A339067.
%K A339303 nonn,tabl
%O A339303 1,4
%A A339303 _Andrew Howroyd_, Dec 04 2020