A038218 Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j)*12^j (with i, j >= 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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• easy

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

Examples

			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)
		

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]

Crossrefs

Cf. A001021 (main diagonal), A001023 (row sums).

Programs

Magma

/* As triangle */ [[Binomial(i,j)*2^(i-j)*12^j: j in [0..i]]: i in [0.. 15]]; // Vincenzo Librandi, Jul 24 2019

Mathematica

Flatten[Table[Binomial[i, j] 2^(i - j) 12^j, {i, 0, 8}, {j, 0, i}]] (* Vincenzo Librandi, Jul 24 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)

Extensions

Name edited by Petros Hadjicostas, Jul 23 2019