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A208333
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Triangle of coefficients of polynomials v(n,x) jointly generated with A208332; see the Formula section.
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3
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1, 0, 4, 0, 2, 10, 0, 2, 6, 28, 0, 2, 6, 24, 76, 0, 2, 6, 28, 80, 208, 0, 2, 6, 32, 100, 264, 568, 0, 2, 6, 36, 120, 360, 840, 1552, 0, 2, 6, 40, 140, 464, 1232, 2624, 4240, 0, 2, 6, 44, 160, 576, 1680, 4128, 8064, 11584, 0, 2, 6, 48, 180, 696, 2184, 5952
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OFFSET
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1,3
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COMMENTS
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As triangle T(n,k) with 0<=k<=n, it is (0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (4, -3/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 28 2012
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LINKS
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Table of n, a(n) for n=1..63.
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FORMULA
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u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=2x*u(n-1,x)+2x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
(START) As triangle with 0<=k<=n :
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - 2*T(n-2,k-1) + 2*T(n-2,k-2) with T(0,0) = 1, T(1,0) = 0, T(1,1) = 4 and T(n,k) = 0 if k<0 or if k>n.
G;f : (1-x+2*y*x)/(1-x-2*y*x+2*y*x^2-2*y^2*x^2).
T(n,n) = A026150(n+1).
Sum_{k, 0<=k<=n} T(n,k) = A003946(n).(END)- Philippe Deléham, Feb 28 2012
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EXAMPLE
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First five rows:
1
0...4
0...2...10
0...2...6...28
0...2...6...24...76
First five polynomials u(n,x):
1, 4x, 2x + 10x^2, 2x + 6x^2 + 28x^3, 2x + 6x^2 + 24x^3 + 76x^4.
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MATHEMATICA
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u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + 2 x*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]
Flatten[%] (* A208332 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208333 *)
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CROSSREFS
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Cf. A003946, A026150, A208332.
Sequence in context: A178903 A058054 A011352 * A208748 A134895 A145877
Adjacent sequences: A208330 A208331 A208332 * A208334 A208335 A208336
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KEYWORD
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nonn,tabl
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AUTHOR
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Clark Kimberling, Feb 26 2012
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STATUS
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approved
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