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A208763
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Triangle of coefficients of polynomials u(n,x) jointly generated with A208764; see the Formula section.
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3
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1, 1, 2, 1, 2, 6, 1, 2, 10, 14, 1, 2, 14, 26, 38, 1, 2, 18, 38, 90, 94, 1, 2, 22, 50, 158, 250, 246, 1, 2, 26, 62, 242, 470, 762, 622, 1, 2, 30, 74, 342, 754, 1614, 2138, 1606, 1, 2, 34, 86, 458, 1102, 2866, 4870, 6170, 4094, 1, 2, 38, 98, 590, 1514, 4582
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OFFSET
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1,3
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COMMENTS
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For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1, -2, 0, 0, 0, 0, 0, 0, 0, ....) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 19 2012
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LINKS
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Table of n, a(n) for n=1..62.
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FORMULA
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u(n,x)=u(n-1,x)+2x*v(n-1,x),
v(n,x)=2x*u(n-1,x)+x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
Contribution from Philippe Deléham, Mar 19 2012. (Start)
G.f. : (1-y*x+2*y*x^2-4*y^2*x^2)/(1-x-y*x+y*x^2-4*y^2*x^2).
T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) + 4*T(n-2,k-2), T(1,0) = 1, T(2,0) = 1, T(2,1) = 2, 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
1...2
1...2...6
1...2...10...14
1...2...14...26...38
First five polynomials u(n,x):
1
1 + 2x
1 + 2x + 6x^2
1 + 2x + 10x^2 + 14x^3
1 + 2x + 14x^2 + 26x^3 + 38x^4
(1, 0, -1, 1, 0, 0, ...) DELTA (0, 2, 1, -2, 0, 0...) begins :
1
1, 0
1, 2, 0
1, 2, 6, 0
1, 2, 10, 14, 0
1, 2, 14, 26, 38, 0
1, 2, 18, 38, 90, 94, 0 . Philippe Deléham, Mar 19 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] + 2 x*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + 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[%] (* A208763 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208764 *)
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CROSSREFS
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Cf. A208764, A208510.
Sequence in context: A208757 A133643 A008305 * A133644 A152431 A143965
Adjacent sequences: A208760 A208761 A208762 * A208764 A208765 A208766
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KEYWORD
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nonn,tabl
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AUTHOR
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Clark Kimberling, Mar 02 2012
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STATUS
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approved
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