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A209755
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Triangle of coefficients of polynomials u(n,x) jointly generated with A209756; see the Formula section.
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3
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1, 1, 2, 2, 4, 3, 3, 7, 8, 5, 4, 11, 17, 17, 8, 5, 16, 31, 41, 33, 13, 6, 22, 51, 83, 91, 63, 21, 7, 29, 78, 150, 205, 195, 117, 34, 8, 37, 113, 250, 406, 483, 403, 214, 55, 9, 46, 157, 392, 734, 1039, 1091, 812, 386, 89, 10, 56, 211, 586, 1239, 2023, 2536
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OFFSET
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1,3
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COMMENTS
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Column 1: 1,2,3,4,5,6,....... A000027
Column 2: 1,2,4,7,11,........ A000124
Column 3: 2,6,13,24,......... A105163
Final row terms: 1,2,3,5,.... A000045 (Fibonacci numbers)
Row sums: 1,3,9,23,57,139,... A133654
Alternating row sums: 1,-1,1,-1,1,-1,1,-1,...; A033999
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*u(n-1,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
2...4....3
3...7....8....5
4...11...11...17...8
First three polynomials u(n,x): 1, 1 + 2x, 2 + 4x + 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] + (x + 1)*v[n - 1, x];
v[n_, x_] := x*u[n - 1, 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}] (* A133654 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A001333 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A033999 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A109613 *)
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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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