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
Examples
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 ... ...
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.
Crossrefs
Programs
-
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 *) -
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
Comments