Find nth Hermite number in C++

In this problem, we are given an integer value N. Our task is to create a program to Find nth Hermite number.

Hermite Number is a number is the value of hermite number when there are 0 arguments.

Nth hermite Number is HN = (-2) * (N - 1) * H(N-2)
The base values are H0 = 1 and H0 = 0.

The hermite sequence is − 1, 0, -2, 0, 12, 0, -120, 0, 1680, 0….

Let’s take an example to understand the problem,

Input

N = 7

Output

0

Input

N = 6

Output

-120

Solution Approach

A simple solution to the problem is using the formula for hermite number. This is done using recursion, we can find N^th Term.

Program to illustrate the working of our solution,

Example

#include <iostream>
using namespace std;
int calcNHermiteNumber(int N) {
   if (N == 0)
      return 1;
   if (N % 2 == 1)
      return 0;
   else
      return -2 * (N - 1) * calcNHermiteNumber(N - 2);
}
int main() {
   int N = 10;
   cout<<"The "<<N<<"th hermite Number is "<<calcNHermiteNumber(N);
   return 0;
}

Output

The 10th hermite Number is -30240

Efficient approach

An efficient approach to solve the problem is by using the formula. We can derive the general formula using the recursive formula.

Here, if the value of N is odd, the hermite number is 0.

If the value of N is even, their will be some value defined by the formula,

HN = ( (-1)(N/2)) * ( 2(N/2) ) * (N-1)!!

The (N-1)!! Is semi-factorial which is calculated as (n-1)*(n-3)*...3*1.

Program to illustrate the working of our solution,

Example

#include <iostream>
#include <math.h>
using namespace std;
int calcSemiFact(int n) {
   int factVal = 1;
   for (int i = 1; i <= n; i = i + 2) {
      factVal *= i;
   }
   return factVal;
}
int calcNHermiteNumber(int n) {
   if (n % 2 == 1)
      return 0;
   int HermiteNumber = (pow(2, n / 2)) * calcSemiFact(n - 1);
   if ((n / 2) % 2 == 1)
      HermiteNumber *= -1;
   return HermiteNumber;
}
int main() {
   int N = 10;
   cout<<"The "<<N<<"th hermite Number is "<<calcNHermiteNumber(N);
   return 0;
}

Output

The 10th hermite Number is -30240