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A208519
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Triangle of coefficients of polynomials v(n,x) jointly generated with A208518; see the Formula section.
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3
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1, 2, 2, 3, 5, 3, 4, 9, 11, 5, 5, 14, 26, 23, 8, 6, 20, 50, 65, 45, 13, 7, 27, 85, 145, 150, 86, 21, 8, 35, 133, 280, 385, 329, 160, 34, 9, 44, 196, 490, 840, 952, 692, 293, 55, 10, 54, 276, 798, 1638, 2310, 2232, 1413, 529, 89, 11, 65, 375, 1230, 2940, 4956
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OFFSET
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1,2
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COMMENTS
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coefficient of x^(n-1): Fibonacci(n+1) = A000045(n+1)
row sums: A002878 (bisection of Lucas sequence)
alternating row sums: A000045(n-2), Fibonacci numbers
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LINKS
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FORMULA
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u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
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EXAMPLE
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First five rows:
1
2...2
3...5....3
4...9....11...5
5...14...26...23...8
First five polynomials v(n,x):
1
2 + 2x
3 + 5x + 3x^2
4 + 9x + 11x^2 + 5x^3
5 + 14x + 26x^2 + 23x^3 + 8x^4
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MATHEMATICA
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u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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