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A318602 Triangle read by rows: T(n,k) is the number of rooted hypertrees on n unlabeled nodes with k edges, (0 <= k < n).

%I A318602 #9 Aug 30 2018 11:07:40
%S A318602 1,0,1,0,1,2,0,1,3,4,0,1,5,10,9,0,1,6,20,30,20,0,1,8,33,77,91,48,0,1,
%T A318602 9,49,152,277,268,115,0,1,11,68,269,655,969,790,286,0,1,12,91,428,
%U A318602 1330,2651,3294,2308,719,0,1,14,116,647,2420,6137,10300,10993,6737,1842
%N A318602 Triangle read by rows: T(n,k) is the number of rooted hypertrees on n unlabeled nodes with k edges, (0 <= k < n).
%C A318602 Equivalently, the number of rooted connected graphs on n unlabeled nodes with k blocks where every block is a complete graph.
%H A318602 Andrew Howroyd, <a href="/A318602/b318602.txt">Table of n, a(n) for n = 1..1275</a>
%e A318602 Triangle begins:
%e A318602   1;
%e A318602   0, 1;
%e A318602   0, 1,  2;
%e A318602   0, 1,  3,  4;
%e A318602   0, 1,  5, 10,   9;
%e A318602   0, 1,  6, 20,  30,  20;
%e A318602   0, 1,  8, 33,  77,  91,  48;
%e A318602   0, 1,  9, 49, 152, 277, 268, 115;
%e A318602   0, 1, 11, 68, 269, 655, 969, 790, 286;
%e A318602   ...
%o A318602 (PARI)
%o A318602 EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i ))-1)}
%o A318602 R(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerMT(y*EulerMT(v)))); [Vecrev(p) | p <- v]}
%o A318602 { my(T=R(10));for(n=1, #T, print(T[n])) }
%Y A318602 Rightmost diagonal is A000081 (rooted trees).
%Y A318602 Row sums are A007563.
%Y A318602 Cf. A318601.
%K A318602 nonn,tabl
%O A318602 1,6
%A A318602 _Andrew Howroyd_, Aug 29 2018