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A319539 Array read by antidiagonals: T(n,k) is the number of binary rooted trees with n leaves of k colors and all non-leaf nodes having out-degree 2. 9
1, 2, 1, 3, 3, 1, 4, 6, 6, 2, 5, 10, 18, 18, 3, 6, 15, 40, 75, 54, 6, 7, 21, 75, 215, 333, 183, 11, 8, 28, 126, 495, 1260, 1620, 636, 23, 9, 36, 196, 987, 3600, 8010, 8208, 2316, 46, 10, 45, 288, 1778, 8568, 28275, 53240, 43188, 8610, 98, 11, 55, 405, 2970, 17934, 80136, 232500, 366680, 232947, 32763, 207 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Not all k colors need to be used. The total number of nodes will be 2n-1.

See table 2.1 in the Johnson reference.

LINKS

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

V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012.

FORMULA

T(1,k) = k.

T(n,k) = (1/2)*([n mod 2 == 0]*T(n/2,k) + Sum_{j=1..n-1} T(j,k)*T(n-j,k)).

G.f. of k-th column R(x) satisfies R(k) = k*x + (R(x)^2 + R(x^2))/2.

EXAMPLE

Array begins:

===========================================================

n\k|  1    2      3       4        5        6         7

---+-------------------------------------------------------

1  |  1    2      3       4        5        6         7 ...

2  |  1    3      6      10       15       21        28 ...

3  |  1    6     18      40       75      126       196 ...

4  |  2   18     75     215      495      987      1778 ...

5  |  3   54    333    1260     3600     8568     17934 ...

6  |  6  183   1620    8010    28275    80136    194628 ...

7  | 11  636   8208   53240   232500   785106   2213036 ...

8  | 23 2316  43188  366680  1979385  7960638  26037431 ...

9  | 46 8610 232947 2590420 17287050 82804806 314260765 ...

...

MAPLE

A:= proc(n, k) option remember; `if`(n<2, k*n, `if`(n::odd, 0,

      (t-> t*(1-t)/2)(A(n/2, k)))+add(A(i, k)*A(n-i, k), i=1..n/2))

    end:

seq(seq(A(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Sep 23 2018

PROG

(PARI) \\ here R(n, k) gives k-th column as a vector.

R(n, k)={my(v=vector(n)); v[1]=k; for(n=2, n, v[n]=sum(j=1, (n-1)\2, v[j]*v[n-j]) + if(n%2, 0, binomial(v[n/2]+1, 2))); v}

{my(T=Mat(vector(8, k, R(8, k)~))); for(n=1, #T~, print(T[n, ]))}

CROSSREFS

Columns 1..5 are A001190, A083563, A220816, A220817, A220818.

Main diagonal is A319580.

Cf. A241555, A319254, A319541.

Sequence in context: A192001 A122176 A159881 * A098546 A126277 A253273

Adjacent sequences:  A319535 A319537 A319538 * A319541 A319542 A319543

KEYWORD

nonn,tabl,look

AUTHOR

Andrew Howroyd, Sep 22 2018

STATUS

approved

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Last modified December 16 19:11 EST 2018. Contains 318188 sequences. (Running on oeis4.)