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A209691
Triangle of coefficients of polynomials u(n,x) jointly generated with A209692; see the Formula section.
3
1, 0, 2, 0, 2, 4, 0, 1, 8, 8, 0, 1, 5, 24, 16, 0, 1, 4, 21, 64, 32, 0, 1, 4, 15, 77, 160, 64, 0, 1, 4, 14, 58, 253, 384, 128, 0, 1, 4, 14, 49, 221, 765, 896, 256, 0, 1, 4, 14, 48, 177, 800, 2173, 2048, 512, 0, 1, 4, 14, 48, 165, 654, 2723, 5885, 4608, 1024, 0, 1, 4
OFFSET
1,3
COMMENTS
Combinatorial limit of rows: A007070.
For a discussion and guide to related arrays, see A208510.
FORMULA
u(n,x)=x*u(n-1,x)+x*v(n-1,x),
v(n,x)=u(n-1,x)+2x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
EXAMPLE
First five rows:
1
0...2
0...2...4
0...1...8...8
0...1...5...24...16
First three polynomials v(n,x): 1, 2x, 2x + 4x^2.
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + 2 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]
Flatten[%] (* A209691 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209692 *)
CROSSREFS
Sequence in context: A136716 A117946 A061930 * A221683 A332760 A199335
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 12 2012
STATUS
approved