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A209572
Triangle of coefficients of polynomials v(n,x) jointly generated with A209571; see the Formula section.
3
1, 1, 3, 1, 3, 5, 1, 3, 11, 7, 1, 3, 11, 29, 9, 1, 3, 11, 41, 61, 11, 1, 3, 11, 41, 129, 111, 13, 1, 3, 11, 41, 153, 339, 183, 15, 1, 3, 11, 41, 153, 523, 771, 281, 17, 1, 3, 11, 41, 153, 571, 1571, 1569, 409, 19, 1, 3, 11, 41, 153, 571, 2035, 4161, 2929, 571, 21
OFFSET
1,3
COMMENTS
Combinatorial limit of row n satisfies linear recurrence
r(n)=4*r(n-1)-r(n-2) with r(1)=1 and r(2)=3. For a
discussion and guide to related arrays, see A208510.
FORMULA
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.
EXAMPLE
First five rows:
1
1...3
1...3...5
1...3...11...7
1...3...11...29...9
First three polynomials v(n,x): 1, 1 + 3x , 1 + 3x + 5x^2.
MATHEMATICA
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]
Flatten[%] (* A209571 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209572 *)
CROSSREFS
Sequence in context: A258208 A210952 A208523 * A134231 A225598 A126637
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 11 2012
STATUS
approved