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A339788 Triangle read by rows: T(n,k) is the number of forests with n unlabeled vertices and maximum vertex degree k, (0 <= k < n).

%I A339788 #13 Feb 16 2025 08:34:01
%S A339788 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 A339788 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 A339788 232,103,41,15,6,2,1,1,6,70,523,561,270,105,41,15,6,2,1
%N A339788 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 A339788 A forest is an acyclic graph.
%C A339788 (The component trees here are not rooted. - _N. J. A. Sloane_, Dec 19 2020)
%H A339788 Andrew Howroyd, <a href="/A339788/b339788.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%H A339788 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MaximumVertexDegree.html">Maximum Vertex Degree</a>
%F A339788 T(n,k) = A144215(n,k) - A144215(n,k-1) for k > 0.
%e A339788 Triangle begins:
%e A339788   1;
%e A339788   1, 1;
%e A339788   1, 1,  1;
%e A339788   1, 2,  2,   1;
%e A339788   1, 2,  4,   2,   1;
%e A339788   1, 3,  7,   6,   2,   1;
%e A339788   1, 3, 11,  13,   6,   2,  1;
%e A339788   1, 4, 17,  30,  15,   6,  2,  1;
%e A339788   1, 4, 25,  60,  39,  15,  6,  2, 1;
%e A339788   1, 5, 36, 128,  94,  41, 15,  6, 2, 1;
%e A339788   1, 5, 50, 254, 232, 103, 41, 15, 6, 2, 1;
%e A339788   ...
%o A339788 (PARI) \\ Here V(n, k) gives column k of A144528.
%o A339788 EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A339788 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 A339788 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 A339788 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 A339788 { my(T=M(12)); for(n=1, #T~, print(T[n, 1..n])) }
%Y A339788 Row sums are A005195.
%Y A339788 Cf. A144215 (max degree <= k), A144528, A238414 (trees), A263293 (graphs).
%K A339788 nonn,tabl
%O A339788 1,8
%A A339788 _Andrew Howroyd_, Dec 18 2020