|
|
A105851
|
|
Binomial transform triangle, read by rows.
|
|
2
|
|
|
1, 2, 1, 4, 3, 1, 8, 8, 4, 1, 16, 20, 12, 5, 1, 32, 48, 32, 16, 6, 1, 64, 112, 80, 44, 20, 7, 1, 128, 256, 192, 112, 56, 24, 8, 1, 256, 576, 448, 272, 144, 68, 28, 9, 1, 512, 1280, 1024, 640, 352, 176, 80, 32, 10, 1, 1024, 2816, 2304, 1472, 832, 432, 208, 92, 36, 11, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
Let P = Pascal's triangle as an infinite lower triangular matrix and A is the infinite array of arithmetic sequences as shown in A077028:
1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, ...
1, 3, 5, 7, 9, ...
1, 4, 7, 10, 13, ...
1, 5, 9, 13, 17, ...
Perform the operation P * A, getting a new array with each column being the binomial transform of an arithmetic sequence. Take antidiagonals of the new array, then by rows = the triangle of A105851.
|
|
LINKS
|
|
|
FORMULA
|
n-th column of the triangle is the binomial transform of the arithmetic sequence (n*k + 1), (k = 0, 1, 2, ...).
T(n,k) = (2 + k*(n - k))*2^(n-k-1) for 0 <= k <= n.
O.g.f.: (1 - x*(2 + t) + 3*t*x^2)/((1 - 2*x)^2*(1 - t*x)^2) = 1 + (2 + t)*x + (4 + 3*t + t^2)*x^2 + ....
k-th column g.f.: (1 + (k - 2)*x)/(1 - 2*x)^2. Cf. A077028. (End)
|
|
EXAMPLE
|
Column 3: 1, 5, 16, 44, 112, ... (A053220) is the binomial transform of 3k+1 (A016777: 1, 4, 7, ...).
Triangle begins:
1;
2, 1;
4, 3, 1;
8, 8, 4, 1;
16, 20, 12, 5, 1;
32, 48, 32, 16, 6, 1;
64, 112, 80, 44, 20, 7, 1;
128, 256, 192, 112, 56, 24, 8, 1;
256, 576, 448, 272, 144, 68, 28, 9, 1;
512, 1280, 1024, 640, 352, 176, 80, 32, 10, 1;
1024, 2816, 2304, 1472, 832, 432, 208, 92, 36, 11, 1;
...
|
|
MAPLE
|
seq(seq((2 + k*(n - k))*2^(n-k-1), k=0..n), n=0..10); # Peter Bala, Jul 26 2015
|
|
MATHEMATICA
|
t[n_, k_]:=(2 + k (n - k)) 2^(n - k - 1); Table[t[n - 1, k - 1], {n, 10}, {k, n}]//Flatten (* Vincenzo Librandi, Jul 26 2015 *)
|
|
PROG
|
(Magma) /* As triangle */ [[(2+k*(n-k))*2^(n-k-1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 26 2015
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|