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A210871
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Triangle of coefficients of polynomials v(n,x) jointly generated with A210870; see the Formula section.
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3
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1, 1, 2, 1, 1, 3, 1, 3, 2, 5, 1, 2, 7, 3, 8, 1, 4, 5, 15, 5, 13, 1, 3, 12, 10, 30, 8, 21, 1, 5, 9, 31, 20, 58, 13, 34, 1, 4, 18, 22, 73, 38, 109, 21, 55, 1, 6, 14, 54, 51, 162, 71, 201, 34, 89, 1, 5, 25, 40, 145, 111, 344, 130, 365, 55, 144, 1, 7, 20, 85, 105, 361, 233
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OFFSET
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1,3
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COMMENTS
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Row n, for n>2, starts with 1 and A028242(n) and ends with F(n-1) and F(n+1), 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)=u(n-1,x)+x*v(n-1,x),
v(n,x)=(x+1)*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 six rows:
1
1...2
1...1...3
1...3...2....5
1...2...7....3....8
1...4...5....15...5...13
First three polynomials v(n,x): 1, 1 + 2x, 1 + x + 3x^2
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MATHEMATICA
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u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := (x + 1)*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]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A000975 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A001045 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A113954 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A077925 *)
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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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