A319590 Number of binary rooted trees with n leaves spanning an initial interval of positive integers and all non-leaf nodes having out-degree 2.

1, 2, 8, 58, 576, 7440, 117628, 2201014, 47552012, 1164812674, 31898271660, 965666303078, 32022547868872, 1154362247246714, 44945574393963472, 1879720975031634318, 84039891496643620196, 3999886612000379135606, 201919706444252727224852, 10775953237291840618917900

Offset: 1

Author

Andrew Howroyd, Sep 23 2018

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200

Crossrefs

Row sums of A319541.

Cf. A316651, A319539.

Programs

Maple

b:= proc(n, k) option remember; `if`(n<2, k*n, `if`(n::odd, 0,
      (t-> t*(1-t)/2)(b(n/2, k)))+add(b(i, k)*b(n-i, k), i=1..n/2))
    end:
a:= n-> add(add((-1)^i*binomial(k, i)*b(n, k-i), i=0..k), k=0..n):
seq(a(n), n=1..23);  # Alois P. Heinz, Sep 07 2019

Mathematica

b[n_, k_] := b[n, k] = If[n < 2, k n, If[OddQ[n], 0, Function[t, t(1 - t)/2][b[n/2, k]]] + Sum[b[i, k] b[n - i, k], {i, 1, n/2}]];
a[n_] := Sum[Sum[(-1)^i Binomial[k, i] b[n, k - i], {i, 0, k}], {k, 0, n}];
Array[a, 23] (* Jean-François Alcover, Apr 10 2020, after Alois P. Heinz *)

PARI

\\ here R(n, k) is k-th column of A319539 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}
seq(n)={sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) )}