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A210869
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Triangle of coefficients of polynomials v(n,x) jointly generated with A210868; see the Formula section.
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3
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1, 0, 2, 1, 0, 3, 0, 3, 0, 5, 1, 0, 7, 0, 8, 0, 4, 0, 15, 0, 13, 1, 0, 12, 0, 30, 0, 21, 0, 5, 0, 31, 0, 58, 0, 34, 1, 0, 18, 0, 73, 0, 109, 0, 55, 0, 6, 0, 54, 0, 162, 0, 201, 0, 89, 1, 0, 25, 0, 145, 0, 344, 0, 365, 0, 144, 0, 7, 0, 85, 0, 361, 0, 707, 0, 655, 0, 233, 1, 0
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OFFSET
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1,3
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COMMENTS
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Row n starts with 1 or 0 and ends with F(n+1), where F=A000045 (Fibonacci numbers).
Row sums: 1,2,4,8,16,32,... (A000079)
Alternating row sums: 1, -2, 4, -8, 16,... (A122803)
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),
where u(1,x)=1, v(1,x)=1.
T(n,k) = T(n-1,k-1) + T(n-2,k) + T(n-2,k-2), T(1,0) = 1, T(2,0) = 0, T(2,1) = 2 and T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Apr 02 2012
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EXAMPLE
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First six rows:
1
0...2
1...0...3
0...3...0...5
1...0...7...0....8
0...4...0...15...0...13
First three polynomials v(n,x): 1, 2x, 1 + 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];
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}] (* A000079 *)
Table[v[n, x]/.x->1, {n, 1, z}] (* A000079 *)
Table[u[n, x]/.x->-1, {n, 1, z}] (* A151575 *)
Table[v[n, x]/.x->-1, {n, 1, z}] (* A122803 *)
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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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