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

0,2

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.

Links

Table of n, a(n) for n=0..54.

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

Example

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.

Sequence in context: A075149 A161618 A202873 * A350394 A013915 A136445

Adjacent sequences: A157930 A157931 A157932 * A157934 A157935 A157936

Keyword

nonn,tabl

Author

M. F. Hasler, Mar 16 2009

Status

approved