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%I #54 May 21 2023 17:11:20
%S 1,1,1,2,1,4,1,6,4,1,8,12,1,10,24,8,1,12,40,32,1,14,60,80,16,1,16,84,
%T 160,80,1,18,112,280,240,32,1,20,144,448,560,192,1,22,180,672,1120,
%U 672,64,1,24,220,960,2016,1792,448,1,26,264,1320,3360,4032,1792,128,1,28
%N Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0 <= k <= floor(n/2)).
%C Row sums are the Jacobsthal numbers (A001045).
%C Apparently, T(n,k)/2^n equals the probability P that n will occur as a partial sum in a randomly-generated infinite sequence of 1s and 2s with n compositions (ordered partitions) into (n-2k) 1s and k 2s. Example: T(6,2)=24; P = 3/8 (24/2^6) that 6 will occur as a partial sum in the sequence with 2 (6-2*2) 1s and 2 2s. - _Bob Selcoe_, Jul 06 2013
%C From _Johannes W. Meijer_, Aug 28 2013: (Start)
%C The antidiagonal sums are A077949 and the backwards antidiagonal sums are A052947.
%C Moving the terms in each column of this triangle, see the example, upwards to row 0 gives the Pell-Jacobsthal triangle A013609 as a square array. (End)
%C The numbers in rows of the triangle are along "first layer" skew diagonals pointing top-right in center-justified triangle given in A013609 ((1+2*x)^n) and along (first layer) skew diagonals pointing top-left in center-justified triangle given in A038207 ((2+x)^n), see links. - _Zagros Lalo_, Jul 31 2018
%C If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 2.000..., when n approaches infinity. - _Zagros Lalo_, Jul 31 2018
%C It appears that the rows of this array are the coefficients of the Jacobsthal polynomials (see MathWorld link). - _Michel Marcus_, Jun 15 2019
%D Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 80-83, 357-358
%H G. C. Greubel, <a href="/A128099/b128099.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%H Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Falcao/falcao5.html">Intrinsic Properties of a Non-Symmetric Number Triangle</a>, J. Int. Seq., Vol. 26 (2023), Article 23.4.8.
%H Richard Fors, <a href="http://www.math.kth.se/xComb/fors.pdf">Independence Complexes of Certain Families of Graphs</a>, Master's thesis in Mathematics at KTH, presented Aug 19 2011.
%H R. J. Mathar, <a href="http://arxiv.org/abs/1609.03964">Tiling n x m rectangles with 1 x 1 and s x s squares</a> arXiv:1609.03964 [math.CO] (2016).
%H Zagros Lalo, <a href="/A128099/a128099.pdf">First layer skew diagonals in center-justified triangle of coefficients in expansion of (1 + 2x)^n</a>
%H Zagros Lalo, <a href="/A128099/a128099_1.pdf">First layer skew diagonals in center-justified triangle of coefficients in expansion of (2 + x)^n</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JacobsthalPolynomial.html">Jacobsthal Polynomial</a>
%F T(n, k) = 2^k*binomial(n-k,k) = 2^k*A011973(n,k).
%F G.f.: 1/(1-z-2*t*z^2).
%F Sum_{k=0..floor(n/2)} k*(T(n,k) = A095977(n-1).
%F From _Johannes W. Meijer_, Aug 28 2013: (Start)
%F T(n, k) = 2*T(n-2, k-1) + T(n-1, k) with T(n, 0) = 1 and T(n, k) = 0 for k < 0 and k > floor(n/2).
%F T(n, k) = A013609(n-k, k), n >= 0 and 0 <= k <= floor(n/2). (End)
%e Triangle starts:
%e 1;
%e 1;
%e 1, 2;
%e 1, 4;
%e 1, 6, 4;
%e 1, 8, 12;
%e 1, 10, 24, 8;
%e 1, 12, 40, 32;
%p T := proc(n,k) if k<=n/2 then 2^k*binomial(n-k,k) else 0 fi end: for n from 0 to 16 do seq(T(n,k),k=0..floor(n/2)) od; # yields sequence in triangular form
%p T := proc(n, k) option remember: if k<0 or k > floor(n/2) then return(0) fi: if k = 0 then return(1) fi: 2*procname(n-2, k-1) + procname(n-1, k): end: seq(seq(T(n, k), k=0..floor(n/2)), n=0..13); # _Johannes W. Meijer_, Aug 28 2013
%t Table[2^k*Binomial[n - k, k] , {n,0,25}, {k,0,Floor[n/2]}] // Flatten (* _G. C. Greubel_, Dec 28 2016 *)
%t t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, t[n - 1, k] + 2 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/2]}] // Flatten (* _Zagros Lalo_, Jul 31 2018 *)
%Y Cf. (Triangle sums) A001045, A095977, A077949, A052947, A113726, A052942, A077909.
%Y Cf. (Similar triangles) A008315, A011973, A102541.
%Y Cf. A013609, A038207
%Y Cf. A207538
%K nonn,tabf
%O 0,4
%A _Emeric Deutsch_, Feb 18 2007
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