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A120429 Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k leaves (i.e., vertices of degree 0; n>=0, k>=1). A ternary tree is a rooted tree in which each vertex has at most three children and each child of a vertex is designated as its left or middle or right child. 3
1, 3, 9, 3, 27, 27, 1, 81, 162, 30, 243, 810, 360, 15, 729, 3645, 2970, 405, 3, 2187, 15309, 19845, 5670, 252, 6561, 61236, 115668, 56700, 6426, 84, 19683, 236196, 612360, 459270, 98658, 4536, 12, 59049, 885735, 3018060, 3214890, 1122660 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Row n has n + 1 - ceiling(n/3) terms.

Row sums yield A001764.

T(n,1) = 3^n = A000244(n).

Sum_{k>=1} k*T(n,k) = binomial(3n,n) = A005809(n).

LINKS

Table of n, a(n) for n=0..41.

FORMULA

T(n,k) = (1/(n+1))*binomial(n+1,k)*Sum_{j=0..n+1-k}3^(n-2k+j+2)*binomial(n+1-k,j)*binomial(j,k-1-j).

G.f. = G = G(t,z) satisfies G = (1+z(G-1+t))^3.

EXAMPLE

T(2,2)=3 because we have (Q,L,M), (Q,L,R) and (Q,M,R), where Q denotes the root and L (M,R) denotes a left (middle, right) child of Q.

Triangle starts:

    1;

    3;

    9,   3;

   27,  27,   1;

   81, 162,  30;

  243, 810, 360,  15;

MAPLE

T:=proc(n, k) if k<=n+1-ceil(n/3) then (1/(n+1))*binomial(n+1, k)*sum(3^(n+j-2*k+2)*binomial(n+1-k, j)*binomial(j, k-1-j), j=0..n+1-k) else 0 fi end: 1; for n from 1 to 11 do seq(T(n, k), k=1..n+1-ceil(n/3)) od; # yields sequence in triangular form

CROSSREFS

Cf. A001764, A000244, A005809, A120981, A120982, A120983.

Sequence in context: A010259 A255583 A339882 * A101431 A120982 A293634

Adjacent sequences:  A120426 A120427 A120428 * A120430 A120431 A120432

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Jul 21 2006

STATUS

approved

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Last modified March 4 01:30 EST 2021. Contains 341773 sequences. (Running on oeis4.)