How to represent a number in the form of A*10-^n? - vb.net

Question

I recently wrote this program in Visul Basic 13.
it searchs for the nth catalan number but after 48 even Decimal type is too short.
Is there any other way to represent them? like in the form of A*10^n?
Public Class Try_Catalan_Number
'Catalan numbers form a sequence of natural numbers that occur in various counting problems,
'often involving recursively defined objects.
Inherits Base_Number
Public Overrides Sub Test()
Dim Return_Catalan_Value As Decimal
If Function_Catalan(Return_Catalan_Value) = False Then
Return_To_Form_Boolean = False
Else
Return_To_Form_Boolean = True
End If
Return_To_Form_Value = Function_Catalan(Return_Catalan_Value)
End Sub
Private Function Function_Catalan(Return_Catalan_Value As Decimal) As Decimal
'We return a Decimal function because catalan numbers can be very big and decimal is the biggest type.
Dim Binomial_Cofficients As Decimal
Dim Result As Decimal
Dim Number_Of_Loops As Integer
Dim tmpNumber As Object
Dim K As Decimal
Dim N As Decimal
If (Number > 48) Then
Return False
Exit Function
End If
'48 is the largest catalan number position which can be displayed...any position above 48 is too big.
tmpNumber = Number - 1
N = 2 * tmpNumber
K = tmpNumber
Result = 1
For Number_Of_Loops = 1 To K
Result = Result * (N - (K - Number_Of_Loops))
Result = Result / Number_Of_Loops
Next Number_Of_Loops
Binomial_Cofficients = Result
tmpNumber = Number - 1
tmpNumber = ((1 / (1 + tmpNumber)) * Binomial_Cofficients)
Return_Catalan_Value = tmpNumber
Return Return_Catalan_Value
End Function
End Class

Answer 1

[I assume by "Visul Basic 13" you mean the VB which is associated with Visual Studio 2013, i.e. VB version 12.0.]
You can use System.Numerics.BigInteger (you'll have to add a reference to System.Numerics):
Imports System.Numerics
Module Module1
Friend Function Factorial(n As Integer) As BigInteger
If n < 2 Then Return 1
If n = 2 Then Return 2
Dim f As BigInteger = BigInteger.Parse("2")
For i = 3 To n
f *= i
Next
Return f
End Function
Friend Function CatalanNumber(n As Integer) As BigInteger
Return Factorial(2 * n) / (Factorial(n + 1) * Factorial(n))
End Function
Sub Main()
For i = 0 To 550
Console.WriteLine(CatalanNumber(i).ToString())
Next
Console.ReadLine()
End Sub
End Module
I did not test to see the maximum Catalan number it can calculate, and I have no inclination to verify the results beyond those shown on the Wikipedia page.
Optimisations are left as an exercise for the reader ;)
Edit: FWIW, I can get it to run a bit faster by using
Function CatalanNumber(n As Integer) As BigInteger
Dim nFactorial = Factorial(n)
Dim twonFactorial = nFactorial
For i = (n + 1) To 2 * n
twonFactorial = BigInteger.Multiply(twonFactorial, i)
Next
Return twonFactorial / (BigInteger.Pow(nFactorial, 2) * (n + 1))
End Function
The speed increase varies from roughly 50% (n=50) to 20% (n=5000). If you're only using the function a few times for fairly small n, there may be little point worrying about it.
Edit2 Re-writing your function a bit to make it easier to read and removing the off-by-one error, we get:
Private Function Function_Catalan(a As Integer) As BigInteger
If a = 0 Then Return 1
Dim binomialCofficient As BigInteger = BigInteger.One
Dim n As Integer = 2 * a
Dim k As Integer = a - 1
For i As Integer = 1 To k
binomialCofficient = binomialCofficient * (n - (k - i)) / i
Next i
Return binomialCofficient / a
End Function

Answer 2

to get this format you could use:
String.Format("{0:E4}", InputNumber)