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A209571
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Triangle of coefficients of polynomials u(n,x) jointly generated with A209572; see the Formula section.
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3
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1, 1, 1, 1, 4, 1, 1, 4, 9, 1, 1, 4, 15, 16, 1, 1, 4, 15, 44, 25, 1, 1, 4, 15, 56, 105, 36, 1, 1, 4, 15, 56, 185, 216, 49, 1, 1, 4, 15, 56, 209, 524, 399, 64, 1, 1, 4, 15, 56, 209, 732, 1295, 680, 81, 1, 1, 4, 15, 56, 209, 780, 2303, 2864, 1089, 100, 1, 1, 4, 15, 56
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OFFSET
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1,5
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COMMENTS
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Penultimate number in row n is (n-1)^2, for n>1.
Combinatorial limit of row n satisfies linear recurrence
r(n)=4*r(n-1)-r(n-2) with r(1)=1 and r(2)=4. 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)+v(n-1,x),
v(n,x)=2x*u(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
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EXAMPLE
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First five rows:
1
1...1
1...4....1
1...4....9....1
1...4....15...16...1
First three polynomials v(n,x): 1, 1 + x, 1 + 4x + x^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] + v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + 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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