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A210221
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Triangle of coefficients of polynomials u(n,x) jointly generated with A210596; see the Formula section.
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4
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1, 2, 3, 2, 5, 4, 4, 8, 10, 8, 8, 13, 20, 24, 16, 16, 21, 40, 52, 56, 32, 32, 34, 76, 116, 128, 128, 64, 64, 55, 142, 240, 312, 304, 288, 128, 128, 89, 260, 488, 688, 800, 704, 640, 256, 256, 144, 470, 964, 1496, 1856, 1984, 1600, 1408, 512, 512, 233, 840
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OFFSET
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1,2
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COMMENTS
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Row sums: even-indexed Fibonacci numbers
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 0, 2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 25 2012
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LINKS
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Table of n, a(n) for n=1..58.
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FORMULA
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u(n,x)=u(n-1,x)+v(n-1,x),
v(n,x)=u(n-1,x)+2x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
Contribution from Philippe Deléham, Mar 25 2012. (Start)
As DELTA-triangle T(n,k) with 0<=k<=n :
G.f.: (1-2*y*x)/(1-x-2*y*x-x^2+2*y*x^2).
T(n,k) = T(n-1,k) + 2*T(n-1,k) + T(n-2,k) - 2*T(n-2,k-1), T(0,0) = T(1,0) = 1, T(2,0) = 2, T(1,1) = T(2,1) = T(2,0) = 0, T(n,k) = 0 if k<0 or if k>=n. (End)
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EXAMPLE
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First five rows:
1
2
3...2
5...4....4
8...10...8...8
First three polynomials u(n,x): 1, 2, 3 + 2x.
(1, 1, -1, 0, 0, 0, ...) DELTA (0, 0, 2, 0, 0, ...) begins :
1
1, 0
2, 0, 0
3, 2, 0, 0
5, 4, 4, 0, 0
8, 10, 8, 8, 0, 0
13, 20, 24, 16, 16, 0, 0 . Philippe Deléham, Mar 25 2012
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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] + v[n - 1, x];
v[n_, 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[%] (* A210221 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A210596 *)
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CROSSREFS
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Cf. A210596, A208510.
Sequence in context: A141658 A089587 A067316 * A216475 A127433 A055573
Adjacent sequences: A210218 A210219 A210220 * A210222 A210223 A210224
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KEYWORD
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nonn,tabf
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AUTHOR
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Clark Kimberling, Mar 24 2012
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STATUS
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approved
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