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A210215
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Triangle of coefficients of polynomials u(n,x) jointly generated with A210216; see the Formula section.
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3
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1, 2, 1, 2, 4, 1, 2, 5, 7, 1, 2, 5, 12, 11, 1, 2, 5, 13, 26, 16, 1, 2, 5, 13, 33, 51, 22, 1, 2, 5, 13, 34, 79, 92, 29, 1, 2, 5, 13, 34, 88, 176, 155, 37, 1, 2, 5, 13, 34, 89, 221, 365, 247, 46, 1, 2, 5, 13, 34, 89, 232, 530, 709, 376, 56, 1, 2, 5, 13, 34, 89, 233, 596
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OFFSET
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1,2
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COMMENTS
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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
2...1
2...4...1
2...5...7....1
2...5...12...11...1
First three polynomials u(n,x): 1, 2 + x, 2 + 4x + x^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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