A220452 - OEIS

%I #27 Aug 13 2022 20:48:59

%S 1,3,9,37,225,1881,19873,251889,3712257,62286625,1171487361,

%T 24402416193,557542291969,13861636770177,372514645389825,

%U 10759590258589441,332386419622387713,10935312198369141249,381705328034883127297,14089260601787531469825,548302210950105933701121

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

%C 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.

%H Felix A. Pahl, <a href="/A220452/b220452.txt">Table of n, a(n) for n = 1..400</a>

%H Mathematics Stack Exchange, <a href="http://math.stackexchange.com/questions/258503">The n immortals problem</a>

%H V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Interesting asymptotic formulas for binomial sums</a>, Jun 09 2013

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

%F 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

%e 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.

%t Table[Sum[Binomial[n,k]*(2*k-3)!!, {k, 1, n}], {n, 1, 20}] (* _Vaclav Kotesovec_, Dec 17 2012 *)

%o (Java)

%o import java.math.BigInteger;

%o public class A220452 {

%o     public static void main (String [] args) {

%o         int max = Integer.parseInt (args [0]);

%o         BigInteger [] doubleFactorials = new BigInteger [max + 1];

%o         BigInteger [] [] binomialCoefficients = new BigInteger [max + 1] [max + 1];

%o         doubleFactorials [0] = BigInteger.ONE;

%o         for (int n = 1;n <= max;n++) {

%o             binomialCoefficients [n] [0] = BigInteger.ONE;

%o             BigInteger sum = BigInteger.ZERO;

%o             for (int k = 1;k <= n;k++) {

%o                 binomialCoefficients [n] [k] = k == n ? BigInteger.ONE : binomialCoefficients [n - 1] [k - 1].add (binomialCoefficients [n - 1] [k]);

%o                 sum = sum.add (binomialCoefficients [n] [k].multiply (doubleFactorials [k - 1]));

%o             }

%o             System.out.println (n + " " + sum);

%o             doubleFactorials [n] = doubleFactorials [n - 1].multiply (BigInteger.valueOf (2 * n - 1));

%o         }

%o     }

%o }

%Y Cf. A001147.

%Y Partial sums of A084262.

%K nonn,easy

%O 1,2

%A _Felix A. Pahl_, Dec 15 2012