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A317495
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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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2
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1, 2, 4, 8, 1, 16, 4, 32, 12, 64, 32, 1, 128, 80, 6, 256, 192, 24, 512, 448, 80, 1, 1024, 1024, 240, 8, 2048, 2304, 672, 40, 4096, 5120, 1792, 160, 1, 8192, 11264, 4608, 560, 10, 16384, 24576, 11520, 1792, 60, 32768, 53248, 28160, 5376, 280, 1, 65536, 114688, 67584, 15360, 1120, 12
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OFFSET
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0,2
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COMMENTS
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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.)
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 A008998 and Pisot sequences E(4,9), P(4,9) when n > 1, see A020708.
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.
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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. 358, 359.
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LINKS
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Table of n, a(n) for n=0..56.
Zagros Lalo, Second layer of skew diagonals in center-justified triangle of coefficients in expansion of (1 + 2x)^n
Zagros Lalo, Second layer of skew diagonals in center-justified triangle of coefficients in expansion of (2 + x)^n
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FORMULA
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T(n,k) = 2^(n - 3k) / ((n - 3k)! k!) * (n - 2k)! where n >= 0 and k = 0..floor(n/3).
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EXAMPLE
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Triangle begins:
1;
2;
4;
8, 1;
16, 4;
32, 12;
64, 32, 1;
128, 80, 6;
256, 192, 24;
512, 448, 80, 1;
1024, 1024, 240, 8;
2048, 2304, 672, 40;
4096, 5120, 1792, 160, 1;
8192, 11264, 4608, 560, 10;
16384, 24576, 11520, 1792, 60;
32768, 53248, 28160, 5376, 280, 1;
65536, 114688, 67584, 15360, 1120, 12;
131072, 245760, 159744, 42240, 4032, 84;
262144, 524288, 372736, 112640, 13440, 448, 1;
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MATHEMATICA
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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[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
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PROG
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(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
(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
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CROSSREFS
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Row sums give A008998, A020708.
Cf. A013609
Cf. A038207
Cf. A317494
Cf. A128099, A207538.
Cf. A000079 (column 0), A001787 (column 1), A001788 (column 2), A001789 (column 3), A003472 (column 4).
Sequence in context: A102256 A000455 A282821 * A317504 A097888 A030275
Adjacent sequences: A317492 A317493 A317494 * A317496 A317497 A317498
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KEYWORD
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tabf,nonn,easy
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AUTHOR
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Zagros Lalo, Jul 30 2018
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STATUS
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approved
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