cp's OEIS Frontend

A318607 Triangle read by rows: T(n,k) is the number of sets of rooted hypertrees on a total of n unlabeled nodes with a total of k edges, (0 <= k < n).

%I A318607 #10 Aug 30 2018 11:07:34
%S A318607 1,1,1,1,2,2,1,3,6,4,1,4,12,16,9,1,5,20,42,46,20,1,6,30,86,145,128,48,
%T A318607 1,7,42,153,353,483,364,115,1,8,56,248,729,1369,1592,1029,286,1,9,72,
%U A318607 376,1345,3236,5150,5151,2930,719,1,10,90,541,2287,6728,13708,18792,16513,8344,1842
%N A318607 Triangle read by rows: T(n,k) is the number of sets of rooted hypertrees on a total of n unlabeled nodes with a total of k edges, (0 <= k < n).
%C A318607 Equivalently, the number of sets of rooted connected graphs on a total of n unlabeled nodes with a total of k blocks where every block is a complete graph.
%C A318607 Bivariate Euler transform of triangle A318602.
%H A318607 Andrew Howroyd, <a href="/A318607/b318607.txt">Table of n, a(n) for n = 1..1275</a>
%e A318607 Triangle begins:
%e A318607   1;
%e A318607   1, 1;
%e A318607   1, 2, 2;
%e A318607   1, 3, 6, 4;
%e A318607   1, 4, 12, 16, 9;
%e A318607   1, 5, 20, 42, 46, 20;
%e A318607   1, 6, 30, 86, 145, 128, 48;
%e A318607   1, 7, 42, 153, 353, 483, 364, 115;
%e A318607   1, 8, 56, 248, 729, 1369, 1592, 1029, 286;
%e A318607   ...
%e A318607 Case n=3: There are 5 sets of rooted graph which are illustrated below (an x marks a root node). These have 0, 1, 1, 2, 2 blocks so row 3 is 1, 2, 2.
%e A318607       x        o        o        o        o
%e A318607               /        / \        \      /
%e A318607     x   x    x   x    x---o    x---o    x---o
%o A318607 (PARI) \\ here EulerMT is Euler transform (bivariate version).
%o A318607 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 A318607 A(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerMT(y*EulerMT(v)))); [Vecrev(p) | p <- EulerMT(v)]}
%o A318607 { my(T=A(10)); for(n=1, #T, print(T[n])) }
%Y A318607 Rightmost diagonal is A000081 (rooted trees).
%Y A318607 Row sums are A035052.
%Y A318607 Cf. A318601, A318602.
%K A318607 nonn,tabl
%O A318607 1,5
%A A318607 _Andrew Howroyd_, Aug 30 2018