|
|
A038218
|
|
Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j)*12^j (with i, j >= 0).
|
|
0
|
|
|
1, 2, 12, 4, 48, 144, 8, 144, 864, 1728, 16, 384, 3456, 13824, 20736, 32, 960, 11520, 69120, 207360, 248832, 64, 2304, 34560, 276480, 1244160, 2985984, 2985984, 128, 5376, 96768, 967680, 5806080, 20901888, 41803776, 35831808
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
Using the transfer matrix method, Cyvin et al. (1996) derive the equation a(x,y)_{i,j} = binomial(i-1, j-1) * x^{i-j} * y^{j-1}. See Eq. (4) on p. 111 of the paper. If we replace i-1 with i, j-1 with j, x with 2, and y with 12, we get the current triangular array. - Petros Hadjicostas, Jul 23 2019
|
|
LINKS
|
|
|
FORMULA
|
Bivariate g.f.: Sum_{i,j >= 0} T(i,j)*x^i*y^j = 1/(1 - 2*x * (1 + 6*y)).
G.f. for row i >= 0: 2^i * (1 + 6*y)^i.
G.f. for column j >= 0: (12*x)^j/(1 - 2*x)^(j+1).
(End)
|
|
EXAMPLE
|
Triangle T(i,j) (with rows i >= 0 and columns j >= 0) begins as follows:
1;
2, 12;
4, 48, 144;
8, 144, 864, 1728;
16, 384, 3456, 13824, 20736;
32, 960, 11520, 69120, 207360, 248832;
64, 2304, 34560, 276480, 1244160, 2985984, 2985984;
128, 5376, 96768, 967680, 5806080, 20901888, 41803776, 35831808;
... (End)
|
|
MATHEMATICA
|
Flatten[Table[Binomial[i, j] 2^(i - j) 12^j, {i, 0, 8}, {j, 0, i}]] (* Vincenzo Librandi, Jul 24 2019 *)
|
|
PROG
|
(Magma) /* As triangle */ [[Binomial(i, j)*2^(i-j)*12^j: j in [0..i]]: i in [0.. 15]]; // Vincenzo Librandi, Jul 24 2019
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|