A318602 - OEIS

%I #9 Aug 30 2018 11:07:40

%S 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 9,49,152,277,268,115,0,1,11,68,269,655,969,790,286,0,1,12,91,428,

%U 1330,2651,3294,2308,719,0,1,14,116,647,2420,6137,10300,10993,6737,1842

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

%C Equivalently, the number of rooted connected graphs on n unlabeled nodes with k blocks where every block is a complete graph.

%H Andrew Howroyd, <a href="/A318602/b318602.txt">Table of n, a(n) for n = 1..1275</a>

%e Triangle begins:

%e   1;

%e   0, 1;

%e   0, 1,  2;

%e   0, 1,  3,  4;

%e   0, 1,  5, 10,   9;

%e   0, 1,  6, 20,  30,  20;

%e   0, 1,  8, 33,  77,  91,  48;

%e   0, 1,  9, 49, 152, 277, 268, 115;

%e   0, 1, 11, 68, 269, 655, 969, 790, 286;

%e   ...

%o (PARI)

%o 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 R(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerMT(y*EulerMT(v)))); [Vecrev(p) | p <- v]}

%o { my(T=R(10));for(n=1, #T, print(T[n])) }

%Y Rightmost diagonal is A000081 (rooted trees).

%Y Row sums are A007563.

%Y Cf. A318601.

%K nonn,tabl

%O 1,6

%A _Andrew Howroyd_, Aug 29 2018