Trinomial expansion

[Arthur Albano]

Let's take the following polynomial (which is closely related to the binomial expansion, and it is in fact a particular case of trinomial expansion):

(1+x+x²)ⁿ

It's expansion gives a series of coefficients, depending solely on n.

if n = 0,            1;
if n = 1,         1  1  1;
if n = 2,      1  2  3  2  1;
if n = 3,   1  3  6  7  6  3 1;
if n = 4, 1 4 10 16 19 16 10 4 1;

and so on.

So, I want to know the sum of coefficients depending on "n" and also the sum of the coefficients just above the the middle position. I was able to create a recursive formula in Microsoft Excel, but I have no clue on how to create an array out of the coefficients to give different answers, sucha as the sum below the middle element or after the middle element, such as:

if n = 0; sum above = 0;
if n = 1; sum above = 1;
if n = 2; sum above = 3;
if n = 3; sum above = 10;
if n = 4; sum above = 31;

and so on...

The sum of the coefficients is simply n³.

The function, I believe, should look like this:

    int SumAbove(int n)
    {
        int array[][];
        int j = 0;
        array[0][0] = 0;
        for(int i=1;i<=n;i++;j++)
          {
              ((i==0)||(i==(2*n))) ? array[i][j]=1 : array[i][j] = array[i][j-1] + array[i][j]+array[i][j+1];
          }
        int sum = 0;
        for(int i=(n+1);i<=(2*n);i++)
            {
                sum =+ array[n][i]
            }
        return(sum);
    }

[fxsaber]

Sum = 3^n.

From here

[Arthur Albano]

fxsaber:

Sum = 3^n.

From here

Yes, Σ = 3ⁿ.

[Arthur Albano]

fxsaber:

Sum = 3^n.

From here

But I am more interested in n, k coefficient and the sum for k<n.

int n = 9;

int a_array[][];

a_array[0][0] = 1;

for(int i=1;i<=n;i++)
  {
   for(int k=0;k<=(2*i);k++)
     {
      a_array[i][k] =
         ((k-2)>=0)?a_array[i-1][k-2]:0 +
         ((k-1)>=0)?a_array[i-1][k-1]:0 +
         (k<=(2*i))?a_array[i-1][k]:0;
     };
  };

Perhaps I am getting closer. Am I?

[Arthur Albano]

[Arthur Albano]

[Arthur Albano]

Arthur Albano:

An absolutely non-parametric study of anomalies that arise from time to time in market trading. There's no such a thing as period, volume or price levels as inputs.

[nevar]

I  saw  another trader trading bimodal - multimodal distributions.But how do  you trade the bimodal - multimodal distributions ?Do you  evaluate  your  trading results on multimodal trinomial distribution criteria? Do you trade  the multimodal trinomial distribution of  price?  etc..

When I google ''multimodal trinomial distribution criteria for  trading'' I can only find few  papers based on option pricing. 

[Arthur Albano]

nevar:

I  saw  another trader trading bimodal - multimodal distributions.But how do  you trade the bimodal - multimodal distributions ?Do you  evaluate  your  trading results on multimodal trinomial distribution criteria? Do you trade  the multimodal trinomial distribution of  price?  etc..

When I google ''multimodal trinomial distribution criteria for  trading'' I can only find few  papers based on option pricing. 

Take a close look at the following book and tell me if there is a support and resistance based on the volumes. Where will the price land?

[lippmaje]

If you want to do this in MQL the problem is the dynamic 2-dimensional array. As far as I know you can only resize one dimension but not two. Let's assume you know the maximum value of n, say MAX_N, that would make things easier.

What you want is the Trinomial Triangle as shown in #4, especially the values of the center row (1,1,3,7,19,...) and the sum thereof. From all what I've read in the article you have to compute the values recursively.

#define MAX_N 10
int Triangle[MAX_N+1][MAX_N+1];

void TriangleInit() {
   Triangle[0][0] = 1;

   for(int n=1;n<=MAX_N;n++) {
      for(int k=0;k<=n;k++) {
         Triangle[n][k] =
            (k-1>=0 ? Triangle[n-1][k-1] : Triangle[n-1][1]) +
            (n-1>=k ? Triangle[n-1][k] : 0) +
            (n-1>=k+1 ? Triangle[n-1][k+1] : 0);
      }
   }
}

int Trinomial(int n,int k) {
   if(k<0) k=-k;
   return (n>=0 && k<=n ? Triangle[n][k] : 0);
}

The rows of the triangle are counted from 0 upwards that is n>=0, and the middle column is denoted by k=0. That means Trinomial(4,0) is expected to return 19 (row 4, middle column.)

The sum of all values above a center value at row n is built by summing up Trinomial(i,0) for i<n.

[lippmaje]

Arthur Albano:

Take a close look at the following book and tell me if there is a support and resistance based on the volumes. Where will the price land?

An order book means nothing. Virtual SL/TP, Iceberg orders, even fake orders, volume that is not executed could mean anything. Why throw heavy computations against an order book?