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A209167
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Triangle of coefficients of polynomials v(n,x) jointly generated with A209166; see the Formula section.
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3
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1, 2, 2, 3, 5, 3, 5, 12, 12, 5, 8, 25, 35, 25, 8, 13, 50, 89, 89, 50, 13, 21, 96, 207, 263, 207, 96, 21, 34, 180, 455, 698, 698, 455, 180, 34, 55, 331, 959, 1719, 2073, 1719, 959, 331, 55, 89, 600, 1959, 4011, 5643, 5643, 4011, 1959, 600, 89, 144, 1075
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OFFSET
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1,2
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COMMENTS
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Row n begins and ends with F(n+1), where F=A000045 (Fibonacci numbers).
Alternating row sums: 1,0,1,0,1,0,1,0,1,0,1,0,1,0,...
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)=u(n-1,x)+(x+1)*v(n-1,x),
v(n,x)=(x+1)*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
3....1
6....4....1
12...14...7....1
24...40...28...8...1
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MATHEMATICA
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First three polynomials v(n, x): 1, 3 + x, 6 + 4x + x^2.
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 + 1)*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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