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A209415
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Triangle of coefficients of polynomials u(n,x) jointly generated with A209416; see the Formula section.
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6
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1, 1, 1, 1, 3, 1, 1, 4, 6, 1, 1, 6, 11, 10, 1, 1, 7, 21, 25, 15, 1, 1, 9, 30, 57, 50, 21, 1, 1, 10, 45, 99, 133, 91, 28, 1, 1, 12, 58, 168, 275, 280, 154, 36, 1, 1, 13, 78, 250, 523, 675, 546, 246, 45, 1, 1, 15, 95, 370, 885, 1433, 1509, 1002, 375, 55, 1, 1, 16, 120
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OFFSET
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1,5
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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, -2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 02 2012
Up to reflection at the vertical axis, the triangle of numbers given here coincides with the triangle given in A208334, i.e. the numbers are the same just read row-wise in the opposite direction. [Christine Bessenrodt, Jul 21 2012]
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LINKS
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Table of n, a(n) for n=1..69.
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FORMULA
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u(n,x)=x*u(n-1,x)+v(n-1,x),
v(n,x)=(x+1)*u(n-1,x)+x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
Contribution from Philippe Deléham, Apr 02 2012: (Start)
As DELTA-triangle T(n,k) with 0<=k<=n :
G.f.: (1+x-2*y*x-2*y*x^2+y^2*x^2)/(1-2*y*x-x^2-y*x^2+y^2*x^2).
T(n,k) = 2*T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and 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...1
1...3....1
1...4....6....1
1...6....11...10...1
First three polynomials v(n,x): 1, 1 + x, 1 + 3x + x^2.
Contribution from Philippe Deléham, Apr 02 2012: (Start)
(1, 0, 1, -2, 0, 0, 0, ...) DELTA (0, 1, 0, 1, 0, 0, 0, ...) begins:
1
1, 0
1, 1, 0
1, 3, 1, 0
1, 4, 6, 1, 0
1, 6, 11, 10, 1, 0
1, 7, 21, 25, 15, 1, 0
1, 9, 30, 57, 50, 21, 1, 0
1, 10, 45, 99, 133, 91, 28, 1, 0 . (End)
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MATHEMATICA
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u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
v[n_, x_] := (x + 1)*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[%] (* A209415 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209416 *)
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CROSSREFS
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Cf. A209416, A208510, A208334.
Sequence in context: A131238 A133380 A105687 * A058879 A208344 A209172
Adjacent sequences: A209412 A209413 A209414 * A209416 A209417 A209418
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KEYWORD
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nonn,tabl
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AUTHOR
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Clark Kimberling, Mar 09 2012
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STATUS
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approved
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