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A208342
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Triangle of coefficients of polynomials u(n,x) jointly generated with A208343; see the Formula section.
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5
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1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 5, 1, 1, 5, 7, 10, 8, 1, 1, 6, 9, 16, 18, 13, 1, 1, 7, 11, 23, 31, 33, 21, 1, 1, 8, 13, 31, 47, 62, 59, 34, 1, 1, 9, 15, 40, 66, 101, 119, 105, 55, 1, 1, 10, 17, 50, 88, 151, 205, 227, 185, 89, 1, 1, 11, 19, 61, 113, 213, 321, 414
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OFFSET
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1,6
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COMMENTS
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coefficient of x^(n-1): A000045(n) (Fibonacci numbers).
n-th row sum: 2^(n-1)
Mirror image of triangle in A053538. - Philippe Deléham, Mar 05 2012
Subtriangle of the triangle T(n,k) given by (1, 0, -1, 1, 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 12 2012
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LINKS
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Table of n, a(n) for n=1..75.
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FORMULA
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u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=x*u(n-1,x)+x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
T(n,k) = A208747(n,k)/2^k.- Philippe Deléham, Mar 05 2012
Contribution from Philippe Deléham, Mar 12 2012 . (Start)
As DELTA-triangle T(n,k) with 0<=k<=n :
G.f.: (1-y*x+y*x^2-y^2*x^2)/(1-x-y*x+t*x^2-y^2*x^2).
T(n,k) = T(n-1,k) + T(n-1,k-1) - 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...1...2
1...1...3...3
1...1...4...5...5
First five polynomials u(n,x): 1, 1 + x, 1 + x + x^2, 1 + x + 3x^2 + 3x^3, 1 + x + 4x^2 + 5x^3 + 5x^4.
(1, 0, -1, 1, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, ...) begins :
1
1, 0
1, 1, 0
1, 1, 2, 0
1, 1, 3, 3, 0
1, 1, 4, 5, 5, 0
1, 1, 5, 7, 10, 8, 0
1, 1, 6, 9, 16, 18, 13, 0
1, 1, 7, 11, 23, 31, 33, 21, 0
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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_] := 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[%] (* A208342 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208343 *)
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CROSSREFS
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Cf. A208343.
Sequence in context: A183456 A183342 A046688 * A157283 A067049 A090641
Adjacent sequences: A208339 A208340 A208341 * A208343 A208344 A208345
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KEYWORD
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nonn,tabl
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AUTHOR
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Clark Kimberling, Feb 25 2012
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STATUS
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approved
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