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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). 2
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, 1, 7, 42, 153, 353, 483, 364, 115, 1, 8, 56, 248, 729, 1369, 1592, 1029, 286, 1, 9, 72, 376, 1345, 3236, 5150, 5151, 2930, 719, 1, 10, 90, 541, 2287, 6728, 13708, 18792, 16513, 8344, 1842 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

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.

Bivariate Euler transform of triangle A318602.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..1275

EXAMPLE

Triangle begins:

  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;

  1, 7, 42, 153, 353, 483, 364, 115;

  1, 8, 56, 248, 729, 1369, 1592, 1029, 286;

  ...

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.

      x        o        o        o        o

              /        / \        \      /

    x   x    x   x    x---o    x---o    x---o

PROG

(PARI) \\ here EulerMT is Euler transform (bivariate version).

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)}

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

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

CROSSREFS

Rightmost diagonal is A000081 (rooted trees).

Row sums are A035052.

Cf. A318601, A318602.

Sequence in context: A046726 A082137 A091187 * A340106 A259824 A065173

Adjacent sequences:  A318604 A318605 A318606 * A318608 A318609 A318610

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Aug 30 2018

STATUS

approved

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Last modified July 25 13:59 EDT 2021. Contains 346290 sequences. (Running on oeis4.)