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A209819
Triangle of coefficients of polynomials u(n,x) jointly generated with A209820; see the Formula section.
3
1, 1, 3, 1, 5, 7, 1, 5, 17, 17, 1, 5, 21, 53, 41, 1, 5, 21, 81, 157, 99, 1, 5, 21, 89, 289, 449, 239, 1, 5, 21, 89, 361, 973, 1253, 577, 1, 5, 21, 89, 377, 1389, 3133, 3433, 1393, 1, 5, 21, 89, 377, 1565, 5085, 9745, 9273, 3363, 1, 5, 21, 89, 377, 1597, 6285
OFFSET
1,3
COMMENTS
Let T(n,k) be the general term.
T(n,n): A001333
T(n,n-1): A088210
Row sums: A003561
Alternating row sums: 1,-2,3,-4,5,-6,7,-8,...
Limiting row: F(2), F(5),F(8),...where F=A000045 (Fibonacci numbers)
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
1...3
1...5...7
1...5...17...17
1...5...21...53...41
First three polynomials u(n,x): 1, 1 + 3x, 1 + 5x + 7x^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: A016600 A130418 A038871 * A193648 A221881 A201811
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 23 2012
STATUS
approved