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
B. N. Cyvin, J. Brunvoll, and S. J. Cyvin, Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.
Gábor Kallós, A generalization of Pascal's triangle using powers of base numbers, Ann. Math. Blaise Pascal 13(1) (2006), 1-15. [See Section 2 of the paper with title "ab-based triangles". Apparently, this is a 2(12)-based triangle; i.e., a = 2 and b = 12 even though b = 12 > 9. - Petros Hadjicostas, Jul 30 2019]
FORMULA
From Petros Hadjicostas, Jul 23 2019: (Start)
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
From Petros Hadjicostas, Jul 23 2019: (Start)
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
Name edited by Petros Hadjicostas, Jul 23 2019
STATUS
approved