A220452 Number of unordered full binary trees with labels from a set of n labels.

1, 3, 9, 37, 225, 1881, 19873, 251889, 3712257, 62286625, 1171487361, 24402416193, 557542291969, 13861636770177, 372514645389825, 10759590258589441, 332386419622387713, 10935312198369141249, 381705328034883127297, 14089260601787531469825, 548302210950105933701121

Offset: 1

Author

Felix A. Pahl, Dec 15 2012

Keywords

• nonn
• easy

Comments

a(n) is the size of the population generated by n unrelated ancestors if two individuals produce one descendant together if and only if they are not related.

Examples

			For n=3, each of the three pairs of ancestors produces one descendant, and each of these descendants produces one more descendant with the respective remaining ancestor; three ancestors, three first-order descendants and three second-order descendants makes a population of a(3)=9.
		

Links

• Felix A. Pahl, Table of n, a(n) for n = 1..400
• Mathematics Stack Exchange, The n immortals problem
• V. Kotesovec, Interesting asymptotic formulas for binomial sums, Jun 09 2013

Crossrefs

Cf. A001147.

Partial sums of A084262.

Programs

Java

import java.math.BigInteger;
public class A220452 {
    public static void main (String [] args) {
        int max = Integer.parseInt (args [0]);
        BigInteger [] doubleFactorials = new BigInteger [max + 1];
        BigInteger [] [] binomialCoefficients = new BigInteger [max + 1] [max + 1];
        doubleFactorials [0] = BigInteger.ONE;
        for (int n = 1;n <= max;n++) {
            binomialCoefficients [n] [0] = BigInteger.ONE;
            BigInteger sum = BigInteger.ZERO;
            for (int k = 1;k <= n;k++) {
                binomialCoefficients [n] [k] = k == n ? BigInteger.ONE : binomialCoefficients [n - 1] [k - 1].add (binomialCoefficients [n - 1] [k]);
                sum = sum.add (binomialCoefficients [n] [k].multiply (doubleFactorials [k - 1]));
            }
            System.out.println (n + " " + sum);
            doubleFactorials [n] = doubleFactorials [n - 1].multiply (BigInteger.valueOf (2 * n - 1));
        }
    }
}

Mathematica

Table[Sum[Binomial[n,k]*(2*k-3)!!, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Dec 17 2012 *)

Formula

a(n) = Sum_{k=1..n} binomial(n,k)*(2k-3)!!.

a(n) ~ (2n-3)!!*sqrt(e) ~ (2n)!/(n!*2^n*(2n-1))*sqrt(e) ~ n^(n-1)*2^(n-1/2)*exp(1/2-n). - Vaclav Kotesovec, Dec 17 2012