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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. 0
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 (list; table; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn,tabl
AUTHOR
M. F. Hasler, Mar 16 2009
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)