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%I #38 Sep 08 2022 08:44:53
%S 1,2,12,4,48,144,8,144,864,1728,16,384,3456,13824,20736,32,960,11520,
%T 69120,207360,248832,64,2304,34560,276480,1244160,2985984,2985984,128,
%U 5376,96768,967680,5806080,20901888,41803776,35831808
%N Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j)*12^j (with i, j >= 0).
%C 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
%H B. N. Cyvin, J. Brunvoll, and S. J. Cyvin, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match34/match34_109-121.pdf">Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons</a>, Match, No. 34 (Oct 1996), pp. 109-121.
%H Gábor Kallós, <a href="http://www.numdam.org/article/AMBP_2006__13_1_1_0.pdf">A generalization of Pascal's triangle using powers of base numbers</a>, 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]
%F From _Petros Hadjicostas_, Jul 23 2019: (Start)
%F Bivariate g.f.: Sum_{i,j >= 0} T(i,j)*x^i*y^j = 1/(1 - 2*x * (1 + 6*y)).
%F G.f. for row i >= 0: 2^i * (1 + 6*y)^i.
%F G.f. for column j >= 0: (12*x)^j/(1 - 2*x)^(j+1).
%F (End)
%e From _Petros Hadjicostas_, Jul 23 2019: (Start)
%e Triangle T(i,j) (with rows i >= 0 and columns j >= 0) begins as follows:
%e 1;
%e 2, 12;
%e 4, 48, 144;
%e 8, 144, 864, 1728;
%e 16, 384, 3456, 13824, 20736;
%e 32, 960, 11520, 69120, 207360, 248832;
%e 64, 2304, 34560, 276480, 1244160, 2985984, 2985984;
%e 128, 5376, 96768, 967680, 5806080, 20901888, 41803776, 35831808;
%e ... (End)
%t Flatten[Table[Binomial[i, j] 2^(i - j) 12^j, {i, 0, 8}, {j, 0, i}]] (* _Vincenzo Librandi_, Jul 24 2019 *)
%o (Magma) /* As triangle */ [[Binomial(i,j)*2^(i-j)*12^j: j in [0..i]]: i in [0.. 15]]; // _Vincenzo Librandi_, Jul 24 2019
%Y Cf. A001021 (main diagonal), A001023 (row sums).
%K nonn,tabl,easy
%O 0,2
%A _N. J. A. Sloane_.
%E Name edited by _Petros Hadjicostas_, Jul 23 2019
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