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A210216
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Triangle of coefficients of polynomials v(n,x) jointly generated with A210215; see the Formula section.
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3
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1, 1, 2, 1, 3, 3, 1, 3, 7, 4, 1, 3, 8, 14, 5, 1, 3, 8, 20, 25, 6, 1, 3, 8, 21, 46, 41, 7, 1, 3, 8, 21, 54, 97, 63, 8, 1, 3, 8, 21, 55, 133, 189, 92, 9, 1, 3, 8, 21, 55, 143, 309, 344, 129, 10, 1, 3, 8, 21, 55, 144, 364, 674, 591, 175, 11, 1, 3, 8, 21, 55, 144, 376, 894
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OFFSET
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1,3
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COMMENTS
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Limiting row: even-indexed Fibonacci numbers, A001906.
n-th row sum: -1+2*n
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)+v(n-1,x)+1,
v(n,x)=xu(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...2
1...3...3
1...3...7...4
1...3...8...14...5
First three polynomials v(n,x): 1, 1 + 2x , 1 + 3x + 3x^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] + v[n - 1, x] + 1;
v[n_, x_] := x*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]
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)
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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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