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%I #25 Sep 08 2022 08:46:22
%S 1,2,4,8,1,16,4,32,12,64,32,1,128,80,6,256,192,24,512,448,80,1,1024,
%T 1024,240,8,2048,2304,672,40,4096,5120,1792,160,1,8192,11264,4608,560,
%U 10,16384,24576,11520,1792,60,32768,53248,28160,5376,280,1,65536,114688,67584,15360,1120,12
%N Triangle read by rows: T(0,0) = 1; T(n,k) =2 * T(n-1,k) + T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.
%C The numbers in rows of the triangle are along a "second layer" of skew diagonals pointing top-left in center-justified triangle given in A013609 ((1+2*x)^n) and along a "second layer" of skew diagonals pointing top-right in center-justified triangle given in A038207 ((2+x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (1+2*x)^n and (2+x)^n are given in A128099 and A207538 respectively.)
%C The coefficients in the expansion of 1/(1-2x-x^3) are given by the sequence generated by the row sums.
%C The row sums give A008998 and Pisot sequences E(4,9), P(4,9) when n > 1, see A020708.
%C If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 2.205569430400..., when n approaches infinity.
%D Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 358, 359.
%H Zagros Lalo, <a href="/A317495/a317495.pdf">Second layer of skew diagonals in center-justified triangle of coefficients in expansion of (1 + 2x)^n</a>
%H Zagros Lalo, <a href="/A317495/a317495_1.pdf">Second layer of skew diagonals in center-justified triangle of coefficients in expansion of (2 + x)^n</a>
%F T(n,k) = 2^(n - 3k) / ((n - 3k)! k!) * (n - 2k)! where n >= 0 and k = 0..floor(n/3).
%e Triangle begins:
%e 1;
%e 2;
%e 4;
%e 8, 1;
%e 16, 4;
%e 32, 12;
%e 64, 32, 1;
%e 128, 80, 6;
%e 256, 192, 24;
%e 512, 448, 80, 1;
%e 1024, 1024, 240, 8;
%e 2048, 2304, 672, 40;
%e 4096, 5120, 1792, 160, 1;
%e 8192, 11264, 4608, 560, 10;
%e 16384, 24576, 11520, 1792, 60;
%e 32768, 53248, 28160, 5376, 280, 1;
%e 65536, 114688, 67584, 15360, 1120, 12;
%e 131072, 245760, 159744, 42240, 4032, 84;
%e 262144, 524288, 372736, 112640, 13440, 448, 1;
%t t[n_, k_] := t[n, k] = 2^(n - 3k)/((n - 3 k)! k!) (n - 2 k)!; Table[t[n, k], {n, 0, 18}, {k, 0, Floor[n/3]} ] // Flatten
%t t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2 t[n - 1, k] + t[n - 3, k - 1]]; Table[t[n, k], {n, 0, 18}, {k, 0, Floor[n/3]}] // Flatten
%o (GAP) Flat(List([0..20],n->List([0..Int(n/3)],k->2^(n-3*k)/(Factorial(n-3*k)*Factorial(k))*Factorial(n-2*k)))); # _Muniru A Asiru_, Jul 31 2018
%o (Magma) /* As triangle */ [[2^(n-3*k)/(Factorial(n-3*k)*Factorial(k))* Factorial(n-2*k): k in [0..Floor(n/3)]]: n in [0.. 15]]; // _Vincenzo Librandi_, Sep 05 2018
%Y Row sums give A008998, A020708.
%Y Cf. A013609
%Y Cf. A038207
%Y Cf. A317494
%Y Cf. A128099, A207538.
%Y Cf. A000079 (column 0), A001787 (column 1), A001788 (column 2), A001789 (column 3), A003472 (column 4).
%K tabf,nonn,easy
%O 0,2
%A _Zagros Lalo_, Jul 30 2018
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