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A209138
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Triangle of coefficients of polynomials v(n,x) jointly generated with A209137; see the Formula section.
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5
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1, 1, 2, 2, 4, 3, 3, 9, 10, 5, 5, 18, 28, 22, 8, 8, 35, 68, 74, 45, 13, 13, 66, 154, 210, 177, 88, 21, 21, 122, 331, 541, 574, 397, 167, 34, 34, 222, 686, 1302, 1656, 1446, 850, 310, 55, 55, 399, 1382, 2982, 4404, 4614, 3434, 1758, 566, 89, 89, 710, 2723
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OFFSET
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1,3
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COMMENTS
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Every row begins and ends with a Fibonacci number (A000045).
u(n,1) = n-th row sum = 3^(n-1).
Alternating row sums: 1,-1,1,-1,1,-1,1,-1,1,-1,...
For a discussion and guide to related arrays, see A208510.
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LINKS
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FORMULA
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u(n,x) = u(n-1,x) + (x+1)*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.
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), T(1,0) = T(2,0) = 1, T(2,1) = 2 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, 2;
2, 4, 3;
3, 9, 10, 5;
5, 18, 28, 22, 8;
First three polynomials v(n,x): 1, 1 + 2x, 2 + 4x + 3x^2.
1;
0, 1;
0, 1, 2;
0, 2, 4, 3;
0, 3, 9, 10, 5;
0, 5, 18, 28, 22, 8;
... (End)
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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] + (x + 1)*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]
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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