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A209820 Triangle of coefficients of polynomials v(n,x) jointly generated with A209819; see the Formula section. 3
1, 2, 2, 2, 6, 5, 2, 8, 18, 12, 2, 8, 30, 52, 29, 2, 8, 34, 104, 146, 70, 2, 8, 34, 136, 342, 402, 169, 2, 8, 34, 144, 514, 1080, 1090, 408, 2, 8, 34, 144, 594, 1848, 3306, 2920, 985, 2, 8, 34, 144, 610, 2360, 6370, 9872, 7746, 2378, 2, 8, 34, 144, 610, 2552 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Let T(n,k) be the general term.
T(n,n): A000129
T(n,n-1): 2*A071667
Row sums: A003462
Alternating row sums: 1,0,1,0,1,0,1,0,...
Limiting row: F(3), F(6),F(9),...where F=A000045 (Fibonacci numbers)
For a discussion and guide to related arrays, see A208510.
LINKS
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...2
2...6...5
2...8...18...12
2...8...30...52...29
First three polynomials v(n,x): 1, 2 + 2x , 2 + 6x + 5x^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] + 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[%] (* A209819 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209820 *)
CROSSREFS
Sequence in context: A309078 A241543 A210740 * A145890 A097091 A094204
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 23 2012
STATUS
approved

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Last modified April 18 03:01 EDT 2024. Contains 371767 sequences. (Running on oeis4.)