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A210753
Triangle of coefficients of polynomials u(n,x) jointly generated with A210754; see the Formula section.
3
1, 2, 2, 3, 7, 4, 4, 16, 20, 8, 5, 30, 61, 52, 16, 6, 50, 146, 198, 128, 32, 7, 77, 301, 575, 584, 304, 64, 8, 112, 560, 1408, 1992, 1616, 704, 128, 9, 156, 966, 3060, 5641, 6328, 4272, 1600, 256, 10, 210, 1572, 6084, 14002, 20330, 18880, 10912, 3584
OFFSET
1,2
COMMENTS
Row n starts with n and ends with 2^(n-1).
Row sums: A007070
Alternating row sums: 1,0,0,0,0,0,0,0,0,....
For a discussion and guide to related arrays, see A208510.
FORMULA
u(n,x)=(x+1)*u(n-1,x)+x*v(n-1,x)+1,
v(n,x)=(x+1)*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
EXAMPLE
First five rows:
1
2...2
3...7....4
4...16...20...8
5...30...61...52...16
First three polynomials u(n,x): 1, 2 + 2x, 3 + 7x + 4x^2.
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := (x + 1)*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*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[%] (* A210753 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A210754 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A007070 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A035344 *)
CROSSREFS
Sequence in context: A222871 A222779 A222894 * A208762 A209746 A354950
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 25 2012
STATUS
approved