%I #12 Jan 10 2021 02:13:49
%S 1,1,1,1,1,1,1,2,2,1,1,2,4,2,1,1,3,7,6,2,1,1,3,11,13,6,2,1,1,4,17,30,
%T 15,6,2,1,1,4,25,60,39,15,6,2,1,1,5,36,128,94,41,15,6,2,1,1,5,50,254,
%U 232,103,41,15,6,2,1,1,6,70,523,561,270,105,41,15,6,2,1
%N Triangle read by rows: T(n,k) is the number of forests with n unlabeled vertices and maximum vertex degree k, (0 <= k < n).
%C A forest is an acyclic graph.
%C (The component trees here are not rooted. - _N. J. A. Sloane_, Dec 19 2020)
%H Andrew Howroyd, <a href="/A339788/b339788.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximumVertexDegree.html">Maximum Vertex Degree</a>
%F T(n,k) = A144215(n,k) - A144215(n,k-1) for k > 0.
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 1, 1;
%e 1, 2, 2, 1;
%e 1, 2, 4, 2, 1;
%e 1, 3, 7, 6, 2, 1;
%e 1, 3, 11, 13, 6, 2, 1;
%e 1, 4, 17, 30, 15, 6, 2, 1;
%e 1, 4, 25, 60, 39, 15, 6, 2, 1;
%e 1, 5, 36, 128, 94, 41, 15, 6, 2, 1;
%e 1, 5, 50, 254, 232, 103, 41, 15, 6, 2, 1;
%e ...
%o (PARI) \\ Here V(n, k) gives column k of A144528.
%o EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o MSet(p,k)={my(n=serprec(p,x)-1); if(min(k,n)<1, 1 + O(x*x^n), polcoef(exp( sum(i=1, min(k,n), (y^i + O(y*y^k))*subst(p + O(x*x^(n\i)), x, x^i)/i ))/(1-y + O(y*y^k)), k, y))}
%o V(n,k)={my(g=1+O(x)); for(n=2, n, g=x*MSet(g, k-1)); Vec(1 + x*MSet(g, k) + (subst(g, x, x^2) - g^2)/2)}
%o M(n, m=n)={my(v=vector(m, k, EulerT(V(n,k-1)[2..1+n])~)); Mat(vector(m, k, v[k]-if(k>1, v[k-1])))}
%o { my(T=M(12)); for(n=1, #T~, print(T[n, 1..n])) }
%Y Row sums are A005195.
%Y Cf. A144215 (max degree <= k), A144528, A238414 (trees), A263293 (graphs).
%K nonn,tabl
%O 1,8
%A _Andrew Howroyd_, Dec 18 2020