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A209140
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Triangle of coefficients of polynomials v(n,x) jointly generated with A209139; see the Formula section.
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3
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1, 1, 3, 2, 5, 7, 3, 12, 18, 17, 5, 23, 51, 58, 41, 8, 45, 118, 189, 175, 99, 13, 84, 264, 506, 645, 507, 239, 21, 155, 558, 1268, 1950, 2085, 1428, 577, 34, 281, 1145, 2974, 5395, 6998, 6482, 3940, 1393, 55, 504, 2286, 6687, 13851, 21141, 23856
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OFFSET
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1,3
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COMMENTS
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Column 1: Fibonacci numbers, A000045.
Alternating row sums: (-2)^(n-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) + 2*T(n-1,k-1) + T(n-2,k) + T(n-2,k-2), T(1,0) = T(2,0) = 1, T(2,1) = 3, T(n,k) = 0 if k < 0 or if k >= n. - Philippe Deléham, Apr 11 2012
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EXAMPLE
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First five rows:
1;
1, 3;
2, 5, 7;
3, 12, 18, 17;
5, 23, 51, 58, 41;
First three polynomials v(n,x):
1
1 + 3x
2 + 5x + 7x^2.
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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] + 2 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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