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A367298
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Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 2 + 4*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 2*x - x^2.
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16
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1, 2, 4, 5, 14, 15, 12, 48, 76, 56, 29, 148, 326, 372, 209, 70, 436, 1212, 1904, 1718, 780, 169, 1242, 4169, 8228, 10191, 7642, 2911, 408, 3456, 13576, 32176, 49992, 51488, 33112, 10864, 985, 9448, 42492, 117304, 218254, 281976, 249612, 140712, 40545, 2378
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OFFSET
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1,2
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COMMENTS
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Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.
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LINKS
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FORMULA
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p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 2 + 4*x, u = p(2,x), and v = 1 - 2*x - x^2.
p(n,x) = k*(b^n - c^n), where k = -(1/sqrt(8 + 8*x + 12*x^2)), b = (1/2)*(4*x + 2 + 1/k), c = (1/2)*(4*x + 2 - 1/k).
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EXAMPLE
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First eight rows:
1
2 4
5 14 15
12 48 76 56
29 148 326 372 209
70 436 1212 1904 1718 780
169 1242 4169 8228 10191 7642 2911
408 3456 13576 32176 49992 51488 33112 10864
Row 4 represents the polynomial p(4,x) = 12 + 48*x + 76*x^2 + 56*x^3, so (T(4,k)) = (12,48,76,56), k=0..3.
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MATHEMATICA
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p[1, x_] := 1; p[2, x_] := 2 + 4 x; u[x_] := p[2, x]; v[x_] := 1 - 2 x - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
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CROSSREFS
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Cf. A000129 (column 1), A001353 (p(n,n-1)), A154244 (row sums, (p(n,1)), A002605 (alternating row sums, (p(n,-1)), A190989 (p(n,2)), A005668 (p(n,-2)), A190869 (p(n,-3)), A094440, A367208, A367209, A367210, A367211, A367297, A367299, A367300, A367301.
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KEYWORD
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AUTHOR
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STATUS
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approved
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