A055327 Triangle of rooted identity trees with n nodes and k leaves.

1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 6, 5, 1, 9, 13, 2, 1, 12, 28, 11, 1, 16, 53, 40, 3, 1, 20, 91, 109, 26, 1, 25, 146, 254, 116, 6, 1, 30, 223, 524, 387, 61, 1, 36, 326, 998, 1068, 329, 12, 1, 42, 461, 1774, 2587, 1289, 145, 1, 49, 634, 2995, 5678, 4133, 911, 25, 1, 56

Offset: 1

Author

Christian G. Bower, May 12 2000

Keywords

• nonn
• tabf
• eigen

Comments

Row lengths are 1,1,1,2,2,3,3,4,4,5,5,6,6,...

Examples

			Triangle begins:
1;
1;
1;
1,  1;
1,  2;
1,  4,  1;
1,  6,  5;
1,  9, 13,  2;
1, 12, 28, 11;
1, 16, 53, 40, 3;
...
From _Joerg Arndt_, Aug 18 2014: (Start)
The identity trees with n=6 nodes, as (preorder-) level sequences, together with their number of leaves, and an ASCII rendering, are:
:
:     1:  [ 0 1 2 3 4 5 ]   1
:  O--o--o--o--o--o
:
:     2:  [ 0 1 2 3 4 3 ]   2
:  O--o--o--o--o
:        .--o
:
:     3:  [ 0 1 2 3 4 2 ]   2
:  O--o--o--o--o
:     .--o
:
:     4:  [ 0 1 2 3 4 1 ]   2
:  O--o--o--o--o
:  .--o
:
:     5:  [ 0 1 2 3 2 1 ]   3
:  O--o--o--o
:     .--o
:  .--o
:
:     6:  [ 0 1 2 3 1 2 ]   2
:  O--o--o--o
:  .--o--o
:
This gives [1, 4, 1], row n=6 of the triangle.
(End)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1226 (first 70 rows)
• N. J. A. Sloane, Transforms
• Index entries for sequences related to rooted trees

Crossrefs

Row sums give A004111.

Columns 2 to 8: A002620(n-2), A055328, A055329, A055330, A055331, A055332, A055333.

A regular version is A301342.

Cf. A055334.

Programs

PARI

WeighMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, (-1)^(i-1)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i ))-1)}
A(n)={my(v=[y]); for(n=2, n, v=concat([y], WeighMT(v))); apply(p->Vecrev(p/y), v)}
{ my(T=A(15)); for(n=1, #T, print(T[n])) } \\ Andrew Howroyd, Aug 28 2018

Formula

G.f. satisfies A(x,y) = x*y + x*WEIGH(A(x,y)) - x. Shifts up under WEIGH transform.