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A210868
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Triangle of coefficients of polynomials u(n,x) jointly generated with A210869; see the Formula section.
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2
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1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 1, 2, 5, 3, 5, 1, 3, 5, 10, 5, 8, 1, 3, 9, 10, 20, 8, 13, 1, 4, 9, 22, 20, 38, 13, 21, 1, 4, 14, 22, 51, 38, 71, 21, 34, 1, 5, 14, 40, 51, 111, 71, 130, 34, 55, 1, 5, 20, 40, 105, 111, 233, 130, 235, 55, 89, 1, 6, 20, 65, 105, 256, 233, 474
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OFFSET
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1,6
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COMMENTS
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In row n the first two terms are 1 and floor(n/2), and the last two, for n>1, are F(n-1) and F(n), where F = A000045 (Fibonacci numbers).
Row sums: 1,2,4,8,16,32,...; A000079.
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 0, -1, 0, 0, 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, Apr 02 2012
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LINKS
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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+1)*u(n-1,x)+(x-1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
As DELTA-triangle T(n,k) with 0<=k<=n :
G.f.: (1+x-y*x-y^2*x^2)/(1-y*x-y^2*x^2-x^2).
T(n,k) = T(n-1,k-1) + T(n-2,k) + 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 six rows:
1
1...1
1...1...2
1...2...2...3
1...2...5...3....5
1...3...5...10...5...8
First three polynomials u(n,x): 1, 1 + x, 1 + x + 2x^2.
(1, 0, -1, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, ...) begins :
1
1, 0
1, 1, 0
1, 1, 2, 0
1, 2, 2, 3, 0
1, 2, 5, 3, 5, 0
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MATHEMATICA
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u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := (x + n)*u[n - 1, x] + x*v[n - 1, x] - 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]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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