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A209130
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Triangle of coefficients of polynomials v(n,x) jointly generated with A102756; see the Formula section.
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3
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1, 1, 2, 1, 5, 3, 1, 9, 12, 5, 1, 14, 31, 27, 8, 1, 20, 65, 89, 55, 13, 1, 27, 120, 230, 222, 108, 21, 1, 35, 203, 511, 684, 514, 205, 34, 1, 44, 322, 1022, 1777, 1834, 1125, 381, 55, 1, 54, 486, 1890, 4095, 5442, 4563, 2367, 696, 89, 1, 65, 705, 3288, 8625
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OFFSET
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1,3
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COMMENTS
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Top edge: (1,2,3,5,8,...) = A000045(n+1), Fibonacci numbers.
Alternating row sums: 1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,...
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle T(n,k) given by (1, 0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 08 2012
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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*u(n-1,x) + (x+1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
As DELTA-triangle T(n,k) with 0 <= k <= n:
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) + T(n-2,k-2), T(0,0) = 1, T(1,0) = 1, T(1,1) = 0 and T(n,k) = 0 if k < 0 or if k > n.
G.f.: (1-x-y*x+y*x^2-y^2*x^2)/(1-(2+y)*x-(y^2-1)*x^2).
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EXAMPLE
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First five rows:
1;
1, 2;
1, 5, 3;
1, 9, 12, 5;
1, 14, 31, 27, 8;
First three polynomials v(n,x):
1
1 + 2x
1 + 5x + 3x^2.
(1, 0, 1/2, 1/2, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, 0...) begins:
1;
1, 0;
1, 2, 0;
1, 5, 3, 0;
1, 9, 12, 5, 0;
1, 14, 31, 27, 8, 0;
1, 20, 65, 89, 55, 13, 0; ...
with row sums 1, 1, 3, 9, 27, 81, 243, 729, ... (powers of 3). (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*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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