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

2,2

Links

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

Formula

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

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 *)

Prog

(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]}

Crossrefs

Column 2 of A319376.

Cf. A000669, A050381.

Sequence in context: A089817 A364460 A006320 * A079738 A127741 A073965

Adjacent sequences: A319374 A319375 A319376 * A319378 A319379 A319380

Keyword

nonn

Author

Andrew Howroyd, Sep 17 2018

Status

approved