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A209745
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Triangle of coefficients of polynomials u(n,x) jointly generated with A209746; see the Formula section.
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3
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1, 1, 2, 2, 5, 4, 3, 12, 16, 8, 5, 25, 49, 44, 16, 8, 50, 127, 166, 112, 32, 13, 96, 301, 513, 504, 272, 64, 21, 180, 670, 1408, 1808, 1424, 640, 128, 34, 331, 1427, 3562, 5641, 5816, 3824, 1472, 256, 55, 600, 2939, 8494, 15981, 20330, 17520, 9888
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OFFSET
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1,3
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COMMENTS
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Row n begins with F(n) and ends with 2^(n-1), where F=A000045 (Fibonacci numbers)
Alternating row sums: 1,-1,1,-1,1,-1,1,-1,...
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ....) DELTA (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 24 2012
Riordan array (1/(1-x-x^2), (2*x+x^2)/(1-x-x^2)). - Philippe Deléham, Mar 24 2012
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LINKS
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FORMULA
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u(n,x)=x*u(n-1,x)+(x+1)*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.
T(n,k) = T(n-1, k) + 2*T(n-1,k-1) + T(n-2,k) + T(n-2,k-1), T(1,0) = T(2,0) = 1, T(2,1) = 2, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 24 2012
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EXAMPLE
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First five rows:
1
1...2
2...5....4
3...12...16...8
5...25...49...44...16
First three polynomials u(n,x): 1, 1 + 2x, 2 + 5x + 4x^2.
(0, 1, 1, -1, 0, 0, 0, ...) DELTA (1, 1, 0, 0, 0, ...) begins :
1
0, 1
0, 1, 2
0, 2, 5, 4
0, 3, 12, 16, 8
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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] + (x + 1)*v[n - 1, x];
v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*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]
Table[Expand[v[n, x]], {n, 1, z}]
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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