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A210876
Triangle of coefficients of polynomials u(n,x) jointly generated with A210877; see the Formula section.
3
1, 2, 1, 1, 5, 1, 1, 4, 9, 1, 1, 3, 12, 14, 1, 1, 3, 9, 29, 20, 1, 1, 3, 8, 27, 60, 27, 1, 1, 3, 8, 22, 74, 111, 35, 1, 1, 3, 8, 21, 63, 181, 189, 44, 1, 1, 3, 8, 21, 56, 178, 399, 302, 54, 1, 1, 3, 8, 21, 55, 154, 474, 806, 459, 65, 1, 1, 3, 8, 21, 55, 145, 430, 1169
OFFSET
1,2
COMMENTS
For n>2, each row begins with 1 and ends with 1. If the term in row n and column k is denoted by U(n,k), then U(n,n-2)=A000096(n-1) and U(n,n-3)=A086274(n-1).
Row sums: A000225 (-1+2^n)
Alternating row sums: A077973
Limiting row: 1,3,8,21,55,..., even-indexed Fibonacci numbers
For a discussion and guide to related arrays, see A208510.
FORMULA
u(n,x)=x*u(n-1,x)+v(n-1,x)+1,
v(n,x)=x*u(n-1,x)+x*v(n-1,x)+x,
where u(1,x)=1, v(1,x)=1.
EXAMPLE
First six rows:
1
2...1
1...5...1
1...4...9....1
1...3...12...14...1
1...3...9....29...20...1
First three polynomials u(n,x): 1, 2 + x, 1 + 5x + x^2.
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + x;
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[%] (* A210876 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A210877 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A077973 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)
CROSSREFS
Sequence in context: A069568 A210545 A141323 * A174785 A356399 A373173
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 30 2012
STATUS
approved