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A209415
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Triangle of coefficients of polynomials u(n,x) jointly generated with A209416; see the Formula section.
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6
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1, 1, 1, 1, 3, 1, 1, 4, 6, 1, 1, 6, 11, 10, 1, 1, 7, 21, 25, 15, 1, 1, 9, 30, 57, 50, 21, 1, 1, 10, 45, 99, 133, 91, 28, 1, 1, 12, 58, 168, 275, 280, 154, 36, 1, 1, 13, 78, 250, 523, 675, 546, 246, 45, 1, 1, 15, 95, 370, 885, 1433, 1509, 1002, 375, 55, 1, 1, 16, 120, 505, 1435, 2718, 3564, 3135, 1749, 550, 66, 1
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OFFSET
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1,5
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COMMENTS
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For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 0, 1, -2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 02 2012
Up to reflection at the vertical axis, the triangle of numbers given here coincides with the triangle given in A208334, i.e., the numbers are the same just read row-wise in the opposite direction. - Christine Bessenrodt, Jul 21 2012
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LINKS
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FORMULA
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u(n,x) = x*u(n-1,x) + 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.
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1 + x - 2*y*x - 2*y*x^2 + y^2*x^2)/(1 - 2*y*x - x^2 - y*x^2 + y^2*x^2).
T(n,k) = 2*T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) - 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 five rows:
1;
1, 1;
1, 3, 1;
1, 4, 6, 1;
1, 6, 11, 10, 1;
First three polynomials v(n,x): 1, 1 + x, 1 + 3x + x^2.
(1, 0, 1, -2, 0, 0, 0, ...) DELTA (0, 1, 0, 1, 0, 0, 0, ...) begins:
1;
1, 0;
1, 1, 0;
1, 3, 1, 0;
1, 4, 6, 1, 0;
1, 6, 11, 10, 1, 0;
1, 7, 21, 25, 15, 1, 0;
1, 9, 30, 57, 50, 21, 1, 0;
1, 10, 45, 99, 133, 91, 28, 1, 0; (End)
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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] + 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]
CoefficientList[CoefficientList[Series[(1 + x - 2*y*x - 2*y*x^2 + y^2*x^2)/(1 - 2*y*x - x^2 - y*x^2 + y^2*x^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Jan 03 2018 *)
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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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