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A209141
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Triangle of coefficients of polynomials u(n,x) jointly generated with A209142; see the Formula section.
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3
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1, 2, 1, 4, 5, 2, 8, 16, 12, 3, 16, 44, 49, 25, 5, 32, 112, 166, 127, 50, 8, 64, 272, 504, 513, 301, 96, 13, 128, 640, 1424, 1808, 1408, 670, 180, 21, 256, 1472, 3824, 5816, 5641, 3562, 1427, 331, 34, 512, 3328, 9888, 17520, 20330, 15981, 8494, 2939
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OFFSET
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1,2
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COMMENTS
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Each row begins with a power of 2 and ends with a Fibonacci number. Alternating row sums: all 1's. For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle T(n,k) given by (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 07 2012
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LINKS
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FORMULA
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u(n,x)=u(n-1,x)+(x+1)*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) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = 1, T(1,0) = 1, T(1,1) = 0, T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, Mar 07 2012
G.f.: -x*y/(-1+x*y+x^2*y^2+2*x+x^2*y). - R. J. Mathar, Aug 12 2015
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EXAMPLE
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First five rows:
1
2....1
4....5....2
8....16...12...3
16...44...49...25...5
First three polynomials u(n,x): 1, 2 + x, 4 + 5x + 2x^2
Triangle (1, 1, 0, 0, 0...) DELTA (0, 1, 1, -1, 0, 0, 0, ...) begins :
1
1, 0
2, 1, 0
4, 5, 2, 0
8, 16, 12, 3, 0
16, 44, 49, 25, 5, 0
32, 112, 166, 127, 50, 8, 0
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MATHEMATICA
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u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*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]
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
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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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