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A368149
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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) = 1 + 2*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - x^2.
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0
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1, 1, 2, 2, 4, 3, 3, 10, 10, 4, 5, 20, 31, 20, 5, 8, 40, 78, 76, 35, 6, 13, 76, 184, 232, 161, 56, 7, 21, 142, 406, 636, 582, 308, 84, 8, 34, 260, 861, 1604, 1831, 1296, 546, 120, 9, 55, 470, 1766, 3820, 5215, 4630, 2640, 912, 165, 10, 89, 840, 3533, 8696
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OFFSET
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1,3
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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) = 1 + 2*x, u = p(2,x), and v = 1 - x^2.
p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 + 4*x), b = (1/2)*(2*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).
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EXAMPLE
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First eight rows:
1
1 2
2 4 3
3 10 10 4
5 20 31 20 5
8 40 78 76 35 6
13 76 184 232 161 56 7
21 142 406 636 582 308 84 8
Row 4 represents the polynomial p(4,x) = 3 + 10*x + 10*x^2 + 4*x^3, so (T(4,k)) = (3,10,10,4), k=0..3.
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MATHEMATICA
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p[1, x_] := 1; p[2, x_] := 1 + 2 x; u[x_] := p[2, x]; v[x_] := 1 - 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. A000045 (column 1); A000027 (p(n,n-1)); A000244 (row sums), (p(n,1)); A033999 (alternating row sums), (p(n,-1)); A116415 (p(n,2)), A000748, (p(n,-2)); A152268, (p(n,3)); A190969, (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150.
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KEYWORD
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AUTHOR
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STATUS
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approved
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