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A317505
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Triangle read by rows: T(0,0) = 1; T(n,k) = - T(n-1,k) - 2 T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.
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1
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1, -1, 1, -1, 2, 1, -4, -1, 6, 1, -8, 4, -1, 10, -12, 1, -12, 24, -1, 14, -40, 8, 1, -16, 60, -32, -1, 18, -84, 80, 1, -20, 112, -160, 16, -1, 22, -144, 280, -80, 1, -24, 180, -448, 240, -1, 26, -220, 672, -560, 32, 1, -28, 264, -960, 1120, -192, -1, 30, -312, 1320, -2016, 672, 1, -32, 364, -1760, 3360, -1792, 64, -1, 34, -420, 2288, -5280, 4032, -448
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OFFSET
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0,5
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COMMENTS
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The numbers in rows of the triangle are along "second layer" skew diagonals pointing top-left in center-justified triangle given in A065109 ((2-x)^n) and along "second layer" skew diagonals pointing top-right 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+x+2x^3) are given by the sequence generated by the row sums (see A077973).
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REFERENCES
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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.
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LINKS
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FORMULA
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T(n,k) = (-1)^(n - 3k) * 2^k / ((n - 3k)! k!) * (n - 2k)! where n is a nonnegative integer and k = 0..floor(n/3).
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EXAMPLE
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Triangle begins:
1;
-1;
1;
-1, 2;
1, -4;
-1, 6;
1, -8, 4;
-1, 10, -12;
1, -12, 24;
-1, 14, -40, 8;
1, -16, 60, -32;
-1, 18, -84, 80;
1, -20, 112, -160, 16;
-1, 22, -144, 280, -80;
1, -24, 180, -448, 240;
-1, 26, -220, 672, -560, 32;
1, -28, 264, -960, 1120, -192;
-1, 30, -312, 1320, -2016, 672;
1, -32, 364, -1760, 3360, -1792, 64;
-1, 34, -420, 2288, -5280, 4032, -448;
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MATHEMATICA
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t[n_, k_] := t[n, k] = (-1)^(n - 3k) * 2^k/((n - 3 k)! k!) * (n - 2 k)!; Table[t[n, k], {n, 0, 19}, {k, 0, Floor[n/3]} ] // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, - t[n - 1, k] + 2 t[n - 3, k - 1]]; Table[t[n, k], {n, 0, 19}, {k, 0, Floor[n/3]}] // Flatten
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CROSSREFS
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KEYWORD
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tabf,sign,easy
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AUTHOR
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STATUS
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approved
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