A319254 Array read by antidiagonals: T(n,k) is the number of series-reduced rooted trees with n leaves of k colors.

1, 2, 1, 3, 3, 2, 4, 6, 10, 5, 5, 10, 28, 40, 12, 6, 15, 60, 156, 170, 33, 7, 21, 110, 430, 948, 785, 90, 8, 28, 182, 965, 3396, 6206, 3770, 261, 9, 36, 280, 1890, 9376, 28818, 42504, 18805, 766, 10, 45, 408, 3360, 21798, 97775, 256172, 301548, 96180, 2312

Offset

1,2

Comments

Not all colors need to be used.

See table 2.3 in the Johnson reference.

Links

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

Virginia Perkins Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. South Carolina, 2012.

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  |   2    10      28       60       110       182        280 ...
4  |   5    40     156      430       965      1890       3360 ...
5  |  12   170     948     3396      9376     21798      44856 ...
6  |  33   785    6206    28818     97775    269675     642124 ...
7  |  90  3770   42504   256172   1068450   3496326    9632960 ...
8  | 261 18805  301548  2357138  12081605  46897359  149491104 ...
9  | 766 96180 2195100 22253672 140160650 645338444 2379859608 ...
...

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(A(i, k)+j-1, j)*b(n-i*j, i-1, k), j=0..n/i)))
    end:
A:= (n, k)-> `if`(n<2, n*k, b(n, n-1, k)):
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Sep 17 2018

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[A[i, k] + j - 1, j] b[n - i j, i - 1, k], {j, 0, n/i}]]];
A[n_, k_] := If[n < 2, n k, b[n, n - 1, k]];
Table[A[n, 1 + d - n], {d, 1, 12}, {n, 1, d}] // Flatten (* Jean-François Alcover, Sep 11 2019, after Alois P. Heinz *)

Prog

(PARI) \\ here R(n, k) gives k'th column as a vector.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k)={my(v=[k]); for(n=2, n, v=concat(v, EulerT(concat(v, [0]))[n])); v}
{my(T=Mat(vector(8, k, R(8, k)~))); for(n=1, #T~, print(T[n, ]))} \\ Andrew Howroyd, Sep 15 2018

Crossrefs

Columns 1..5 are A000669, A050381, A220823, A220824, A220825.

Main diagonal is A319369.

Cf. A141610, A242249, A255517, A256064, A256068, A319376.

Sequence in context: A094435 A133341 A111492 * A210864 A349550 A144305

Adjacent sequences: A319251 A319252 A319253 * A319255 A319256 A319257

Keyword

nonn,tabl

Author

Andrew Howroyd, Sep 15 2018

Status

approved