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A209773
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Triangle of coefficients of polynomials u(n,x) jointly generated with A209774; see the Formula section.
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4
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1, 1, 2, 2, 6, 5, 2, 11, 21, 13, 3, 17, 48, 67, 34, 3, 25, 92, 188, 206, 89, 4, 33, 154, 422, 684, 619, 233, 4, 44, 238, 809, 1756, 2365, 1829, 610, 5, 54, 348, 1411, 3801, 6833, 7882, 5334, 1597, 5, 68, 484, 2285, 7369, 16471, 25302, 25549, 15393
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OFFSET
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1,3
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COMMENTS
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Last term in row n: F(2n+1), where F=A000045, the Fibonacci numbers
For a discussion and guide to related arrays, see A208510.
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LINKS
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FORMULA
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u(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x),
v(n,x)=(x+1)*u(n-1,x)+2x*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
1...2
2...6....5
2...11...21...13
3...17...48...67...34
First three polynomials u(n,x): 1, 1 + 2x, 2 + 6x + 5x^2.
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MATHEMATICA
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u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*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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