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A210561
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Triangle of coefficients of polynomials u(n,x) jointly generated with A210562; see the Formula section.
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2
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1, 1, 2, 1, 3, 4, 1, 3, 8, 8, 1, 3, 9, 20, 16, 1, 3, 9, 26, 48, 32, 1, 3, 9, 27, 72, 112, 64, 1, 3, 9, 27, 80, 192, 256, 128, 1, 3, 9, 27, 81, 232, 496, 576, 256, 1, 3, 9, 27, 81, 242, 656, 1248, 1280, 512, 1, 3, 9, 27, 81, 243, 716, 1808, 3072, 2816, 1024, 1, 3, 9
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OFFSET
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1,3
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COMMENTS
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Last term in row n: 2^(n-1)
Limiting row: 3^(k-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)=x*u(n-1,x)+x*v(n-1,x)+1,
v(n,x)=(x+1)*u(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
T(n,k) = 2*T(n-1,k-1) + T(n-2,k-1).
E.g.f. for n-th subdiagonal: exp(2*x)*(1 + x + x^2/2! + x^3/3! + ... + x^n/n!). Cf. A004070.
Riordan array (1/(1 - x), x*(2 + x)).
(End)
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EXAMPLE
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First five rows:
1
1...2
1...3...4
1...3...8...8
1...3...9...20...16
First three polynomials u(n,x): 1, 1 + 2x, 1 + 3x + 4x^2.
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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] + x*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, x] + 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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