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A209759
Triangle of coefficients of polynomials u(n,x) jointly generated with A209760; see the Formula section.
3
1, 1, 2, 1, 5, 5, 1, 5, 16, 13, 1, 5, 19, 48, 34, 1, 5, 19, 68, 141, 89, 1, 5, 19, 71, 233, 409, 233, 1, 5, 19, 71, 262, 772, 1175, 610, 1, 5, 19, 71, 265, 948, 2492, 3349, 1597, 1, 5, 19, 71, 265, 986, 3354, 7879, 9482, 4181, 1, 5, 19, 71, 265, 989, 3641
OFFSET
1,3
COMMENTS
Limiting row: A001834
Coefficient of x^n in u(n,x): odd-indexed Fibonacci numbers
Alternating row sums: 1,-1,1,-1,1,-1,1,-1,...; A033999
For a discussion and guide to related arrays, see A208510.
FORMULA
u(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x),
v(n,x)=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
1...2
1...5...5
1...5...16...13
1...5...19...48...34
First three polynomials u(n,x): 1, 1 + 2x, 1 + 5x + 5x^2.
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := 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[%] (* A209759 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209760 *)
CROSSREFS
Sequence in context: A300661 A145882 A209765 * A111785 A304462 A021468
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 14 2012
STATUS
approved