

A214568


Triangle read by rows: T(n,k) is the number of rooted trees t with n vertices yielding k distinct rooted trees with n+1 vertices when a pendant edge is added to a vertex of t (1 <= k <= n).


4



1, 0, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 2, 3, 3, 0, 1, 1, 6, 6, 6, 0, 1, 3, 7, 11, 14, 12, 0, 1, 1, 11, 16, 29, 32, 25, 0, 1, 3, 11, 26, 46, 72, 75, 52, 0, 1, 2, 16, 27, 79, 122, 182, 177, 113, 0, 1, 3, 18, 42, 101, 217, 336, 457, 420, 247, 0, 1, 1, 20, 47, 149, 303, 621, 911, 1160, 1005, 548
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OFFSET

1,10


COMMENTS

Row n contains n entries.
Sum_{k=1..n} T(n,k) = A000081(n) = number of rooted trees with n vertices.
Sum_{k=1..n} k*T(n,k) = A000107(n).
T(n,n) = A004111(n).
T(n,3) = A032741(n1) = number of proper divisors of n1; if d is a proper divisor of n1 (= number of edges), consider d identical rooted trees with (n1)/d edges, root degree 1, height 2 and identify their roots.
The bivariate g.f. can be computed with eq. (4.2) of HararyRobinson.  R. J. Mathar, Sep 16 2015


LINKS

Sean A. Irvine, Rows n=1..44 of triangle, flattened
F. Harary, R. W. Robinson, Isomorphic factorizations VIII: bisectable trees, Combinatorica 4 (2) (1984) 169179.


FORMULA

No formula available. Entries have been obtained by counting (using Maple) the rooted trees (identified by their MatulaGoebel numbers) with the required properties (using A061775 and A214567).
Bivariate g.f. T(x,y) = x * y * Product_{p>=1} Product_{k=1..p} (1 + x^p*y^k / (1x^p))^(a(p,k)), where a(p,k) is the coefficient of x^p*y^k in T(x,y) [(4.2) from Harari and Robinson]. This allows incremental computation of the rows of the sequence by starting with T(x,y) = x*y (p=1) and increasing p by 1 for each row.  Sean A. Irvine, Oct 10 2017


EXAMPLE

Triangle starts:
1;
0, 1;
0, 1, 1;
0, 1, 1, 2;
0, 1, 2, 3, 3;
0, 1, 1, 6, 6, 6;
0, 1, 3, 7, 11, 14, 12;
0, 1, 1, 11, 16, 29, 32, 25;
Row 4 is 0,1,1,2 because the four rooted trees with 4 vertices generate 2,3,4,and 4 rooted trees with 5 vertices.


CROSSREFS

Cf. A000081, A000107, A004111, A032741, A061775, A214567.
Sequence in context: A145201 A284265 A119464 * A107017 A201076 A318601
Adjacent sequences: A214565 A214566 A214567 * A214569 A214570 A214571


KEYWORD

nonn,tabl


AUTHOR

Emeric Deutsch, Jul 26 2012


STATUS

approved



