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A210794
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Triangle of coefficients of polynomials v(n,x) jointly generated with A210793; see the Formula section.
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3
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1, 1, 2, 3, 3, 3, 3, 11, 8, 5, 9, 18, 29, 17, 8, 9, 48, 67, 71, 35, 13, 27, 81, 180, 194, 158, 68, 21, 27, 189, 387, 575, 508, 338, 129, 34, 81, 324, 918, 1410, 1617, 1222, 695, 239, 55, 81, 702, 1890, 3606, 4471, 4222, 2793, 1393, 436, 89, 243, 1215
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OFFSET
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1,3
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COMMENTS
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Row n starts with a power of 3 and ends with F(n+1), where F=A000045 (Fibonacci numbers).
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+2)*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-1) + 3*T(n-2,k) + 2*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. - Philippe Deléham, Mar 29 2012
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EXAMPLE
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First five rows:
1
1...2
3...3....3
3...11...8....5
9...18...29...17...8
First three polynomials v(n,x): 1, 1 + 2x, 3 + 3x + 3x^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 + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 1; c = 0; h = 2; p = -1; f = 0;
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]
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A000244 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A000244 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A000012 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A077925 *)
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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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