A157933 Triangle T(i,j) such that Sum_{j=0..i} T(i,j)*x(i,j)/2^i = Sum_{k=0..i, j=0..k} x(k,j), if x(k-1,j) = (x(k,j) + x(k,j+1))/2.

1, 3, 3, 7, 10, 7, 15, 25, 25, 15, 31, 56, 66, 56, 31, 63, 119, 154, 154, 119, 63, 127, 246, 337, 372, 337, 246, 127, 255, 501, 711, 837, 837, 711, 501, 255, 511, 1012, 1468, 1804, 1930, 1804, 1468, 1012, 511, 1023, 2035, 2992, 3784, 4246, 4246, 3784, 2992, 2035, 1023

Offset: 0

Author

M. F. Hasler, Mar 16 2009

Keywords

• nonn
• tabl

Comments

Rows and columns are numbered starting with 0. Consider a pyramid (triangle) where each element is the mean value of the two elements below. Then the sum of all elements is expressed as linear combination of the elements at the base. This sequence gives the coefficients times the necessary power of 2.

Examples

			To get the 3rd row of the triangle, consider the pyramid
    f
   d e
  a b c
where d=(a+b)/2, e=(b+c)/2, f=(d+e)/2. Then a+b+c+d+e+f=(7a+10b+7c)/2^2, which yields the row (7,10,7).
Triangle begins:
   1,
   3,   3;
   7,  10,   7;
  15,  25,  25,  15;
  31,  56,  66,  56,  31;
  63, 119, 154, 154, 119, 63;
  ...
		

Crossrefs

Row sums give A001788(n+1).

T(2n,n) gives A033504.

Formula

The first and last term in the (i+1)-th row is T(i,0) = 2^(i+1)-1.

The second and penultimate term is T(i,1) = T(i,0) + T(i-1,1).

G.f.: 1/((1-2*x)*(1-2*x*y)*(1-x-x*y)). - Yu-Sheng Chang, Sep 20 2023