The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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 Cf. A001021 (main diagonal), A001023 (row sums). Sequence in context: A326125 A066700 A159323 * A264841 A191249 A005760 Adjacent sequences:  A038215 A038216 A038217 * A038219 A038220 A038221 KEYWORD nonn,tabl,easy AUTHOR EXTENSIONS Name edited by Petros Hadjicostas, Jul 23 2019 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified February 22 23:44 EST 2020. Contains 332157 sequences. (Running on oeis4.)