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A209998
Triangle of coefficients of polynomials v(n,x) jointly generated with A209996; see the Formula section.
3
1, 2, 3, 2, 8, 9, 2, 10, 30, 27, 2, 10, 46, 108, 81, 2, 10, 50, 198, 378, 243, 2, 10, 50, 242, 810, 1296, 729, 2, 10, 50, 250, 1122, 3186, 4374, 2187, 2, 10, 50, 250, 1234, 4986, 12150, 14580, 6561, 2, 10, 50, 250, 1250, 5946, 21330, 45198, 48114, 19683
OFFSET
1,2
COMMENTS
Row n starts 2, 2*5, 2*5^2,... ; ends with 3^(n-1).
Conjecture: penultimate term in row n is A199923(n).
Alternating row sums: A077925
For a discussion and guide to related arrays, see A208510.
FORMULA
u(n,x)=x*u(n-1,x)+2x*v(n-1,x)+1,
v(n,x)=(x+1)*u(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
EXAMPLE
First five rows:
1
2...3
2...8....9
2...10...30...27
2...10...46...108...81
First three polynomials v(n,x): 1, 2 + 3x , 2 + 8x + 9x^2.
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*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[%] (* A209996 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209998 *)
CROSSREFS
Sequence in context: A134347 A057761 A321477 * A349972 A163204 A364895
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 23 2012
EXTENSIONS
a(55) corrected by Georg Fischer, Sep 03 2021
STATUS
approved