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A209149
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Triangle of coefficients of polynomials v(n,x) jointly generated with A209146; see the Formula section.
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5
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1, 3, 1, 6, 5, 1, 12, 16, 7, 1, 24, 44, 30, 9, 1, 48, 112, 104, 48, 11, 1, 96, 272, 320, 200, 70, 13, 1, 192, 640, 912, 720, 340, 96, 15, 1, 384, 1472, 2464, 2352, 1400, 532, 126, 17, 1, 768, 3328, 6400, 7168, 5152, 2464, 784, 160, 19, 1, 1536, 7424
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OFFSET
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1,2
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COMMENTS
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Alternating row sums: 1,2,2,2,2,2,2,2,2,2,2,2,2,...
For a discussion and guide to related arrays, see A208510.
As triangle T(n,k) with 0 <= k <= n, it is (3, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 08 2012
Riordan array ( (1 + x)/(1 - 2*x), x/(1 - 2*x) ). Cf. A118800. Matrix inverse is a signed version of A112626. - Peter Bala, Jul 17 2013
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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) = u(n-1,x) + (x+1)*v(n-1,x) + 1,
where u(1,x)=1, v(1,x)=1.
As DELTA-triangle:
T(n,k) = 2*T(n-1,k) + T(n-1,k-1), T(0,0) = 1, T(1,0) = 3, T(1,1) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Mar 08 2012
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EXAMPLE
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First five rows:
1;
3, 1;
6, 5, 1;
12, 16, 7, 1;
24, 44, 30, 9, 1;
First three polynomials v(n,x): 1, 3 + x, 6 + 5x + x^2.
v(1,x) = 1
v(2,x) = 3 + x
v(3,x) = (3 + x)*(2 + x)
v(4,x) = (3 + x)*(2 + x)^2
v(5,x) = (3 + x)*(2 + x)^3
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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_] := u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
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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