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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.
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%I #8 Aug 23 2018 16:30:24

%S 1,2,4,8,-1,16,-4,32,-12,64,-32,1,128,-80,6,256,-192,24,512,-448,80,

%T -1,1024,-1024,240,-8,2048,-2304,672,-40,4096,-5120,1792,-160,1,8192,

%U -11264,4608,-560,10,16384,-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 "second layer" skew diagonals pointing top-right in center-justified triangle given in A065109 ((2-x)^n) and along "second layer" skew diagonals pointing top-left in center-justified triangle given in A303872 ((-1+2x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (2-x)^n and (-1+2x)^n are given in A133156 (coefficients of Chebyshev polynomials of the second kind) and A305098 respectively.) The coefficients in the expansion of 1/(1-2x+x^3) are given by the sequence generated by the row sums. The row sums give A000071 (Fibonacci numbers - 1). If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 1.61803398874989484... (A001622: Decimal expansion of Golden ratio (phi or tau) = (1 + sqrt(5))/2), 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. 139-141, 391-393.

%H Shara Lalo, <a href="/A317504/a317504.pdf">Second layer skew diagonals in center-justified triangle of coefficients in expansion of (2 - x)^n</a>

%H Shara Lalo, <a href="/A317504/a317504_1.pdf">Second layer skew diagonals in center-justified triangle of coefficients in expansion of (-1 + 2x)^n</a>

%F T(n,k) = 2^(n - 3k) * (-2)^k / ((n - 3k)! k!) * (n - 2k)! where n is a nonnegative integer 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) * (-1)^k/((n - 3 k)! k!) * (n - 2 k)!; Table[t[n, k], {n, 0, 16}, {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, 16}, {k, 0, Floor[n/3]}] // Flatten

%Y Row sums give A000071.

%Y Cf. A065109, A303872.

%Y Cf. A133156, A305098.

%Y Cf. A001622.

%K tabf,sign,easy

%O 0,2

%A _Shara Lalo_, Aug 02 2018