A319377 Number of series-reduced rooted trees with n leaves of exactly two colors.

1, 6, 30, 146, 719, 3590, 18283, 94648, 497757, 2652898, 14307845, 77958746, 428588051, 2374676854, 13247984959, 74357762790, 419604029622, 2379243477538, 13549087798391, 77458553063930, 444383895880897, 2557639072274418, 14763596994726379, 85449948037167684

Offset: 2

Author

Andrew Howroyd, Sep 17 2018

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 2..200

Crossrefs

Column 2 of A319376.

Cf. A000669, A050381.

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)):
a:= n-> A(n, 2) -2*A(n, 1):
seq(a(n), n=2..30);  # Alois P. Heinz, Sep 18 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]];
a[n_] := A[n, 2] - 2*A[n, 1];
a /@ Range[2, 30] (* Jean-François Alcover, Sep 24 2019, after Alois P. Heinz *)

PARI

\\ here R(n,k) is k-th column of A319254.
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}
seq(n)={(R(n,2)-2*R(n,1))[2..n]}

Formula

a(n) = A050381(n) - 2*A000669(n).