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A209689
Triangle of coefficients of polynomials u(n,x) jointly generated with A209690; see the Formula section.
3
1, 0, 2, 0, 2, 3, 0, 1, 6, 4, 0, 1, 4, 13, 5, 0, 1, 3, 13, 24, 6, 0, 1, 3, 9, 35, 40, 7, 0, 1, 3, 8, 28, 81, 62, 8, 0, 1, 3, 8, 22, 82, 167, 91, 9, 0, 1, 3, 8, 21, 64, 217, 315, 128, 10, 0, 1, 3, 8, 21, 56, 188, 519, 554, 174, 11, 0, 1, 3, 8, 21, 55, 155, 529, 1136, 921
OFFSET
1,3
COMMENTS
Combinatorial limit of rows: even-indexed Fibonacci numbers. 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)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
EXAMPLE
First five rows:
1
0...2
0...2...3
0...1...6...4
0...1...4...13...5
First three polynomials v(n,x): 1, 2x, 2x + 3x^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] + 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[%] (* A209689 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209690 *)
CROSSREFS
Sequence in context: A323474 A132814 A058623 * A204329 A111565 A299761
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 12 2012
STATUS
approved