|
|
A118416
|
|
Triangle read by rows: T(n,k) = (2*k-1)*2^(n-1), 0 < k <= n.
|
|
16
|
|
|
1, 2, 6, 4, 12, 20, 8, 24, 40, 56, 16, 48, 80, 112, 144, 32, 96, 160, 224, 288, 352, 64, 192, 320, 448, 576, 704, 832, 128, 384, 640, 896, 1152, 1408, 1664, 1920, 256, 768, 1280, 1792, 2304, 2816, 3328, 3840, 4352, 512, 1536, 2560, 3584, 4608, 5632, 6656, 7680
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
The alternating row sums, Sum_{k=1..n} (-1)^(k+1)*T(n,k), are: (a) in odd rows, the central term, T(n,(n+1)/2) = A058962((n-1)/2); (b) in even rows, the negation of the average of the two central terms, -(T(2n,n) + T(2n,+1))/2 = -A018215(m/2). The absolute values of the alternating row sums give the plain row means, Sum_{k=1..n} T(n,k)/n; the alternating sign row means are (-2)^(n-1). - Gregory Gerard Wojnar, Feb 10 2024
|
|
LINKS
|
|
|
FORMULA
|
T(n,k) = 2*T(n-1,k), 1 <= k < n; T(n,n) = A014480(n-1).
|
|
EXAMPLE
|
Triangle begins:
1;
2, 6;
4, 12, 20;
8, 24, 40, 56;
16, 48, 80, 112, 144;
32, 96, 160, 224, 288, 352;
64, 192, 320, 448, 576, 704, 832;
|
|
MAPLE
|
|
|
MATHEMATICA
|
Flatten[Table[(2k-1)2^(n-1), {n, 10}, {k, n}]] (* Harvey P. Dale, Aug 26 2014 *)
|
|
PROG
|
(Haskell)
a118416 n k = a118416_tabl !! (n-1) !! (k-1)
a118416_row 1 = [1]
a118416_row n = (map (* 2) $ a118416_row (n-1)) ++ [a014480 (n-1)]
a118416_tabl = map a118416_row [1..]
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|