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A209819
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Triangle of coefficients of polynomials u(n,x) jointly generated with A209820; see the Formula section.
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3
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1, 1, 3, 1, 5, 7, 1, 5, 17, 17, 1, 5, 21, 53, 41, 1, 5, 21, 81, 157, 99, 1, 5, 21, 89, 289, 449, 239, 1, 5, 21, 89, 361, 973, 1253, 577, 1, 5, 21, 89, 377, 1389, 3133, 3433, 1393, 1, 5, 21, 89, 377, 1565, 5085, 9745, 9273, 3363, 1, 5, 21, 89, 377, 1597, 6285
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OFFSET
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1,3
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COMMENTS
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Let T(n,k) be the general term.
Alternating row sums: 1,-2,3,-4,5,-6,7,-8,...
Limiting row: F(2), F(5),F(8),...where F=A000045 (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)+2x*v(n-1,x)+1,
v(n,x)=(x+1)*u(n-1,x)+x*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...3
1...5...7
1...5...17...17
1...5...21...53...41
First three polynomials u(n,x): 1, 1 + 3x, 1 + 5x + 7x^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] + 2 x*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, x] + 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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