A209996 Triangle of coefficients of polynomials u(n,x) jointly generated with A209998; see the Formula section.

1, 1, 3, 1, 5, 9, 1, 5, 21, 27, 1, 5, 25, 81, 81, 1, 5, 25, 117, 297, 243, 1, 5, 25, 125, 513, 1053, 729, 1, 5, 25, 125, 609, 2133, 3645, 2187, 1, 5, 25, 125, 625, 2853, 8505, 12393, 6561, 1, 5, 25, 125, 625, 3093, 12825, 32805, 41553, 19683, 1, 5, 25, 125

Offset

1,3

Comments

Row n starts with 1, 5, 5^2, 5^3,...,5^floor[(n+1)/2] and ends with 3^(n-1).

Denoting the general term by T(n,k), we have T(n,n-1)=A081038.

Alternating row sums: A000975 (signed).

For a discussion and guide to related arrays, see A208510.

Links

Table of n, a(n) for n=1..59.

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...9
1...5...21...27
1...5...25...81...81
First three polynomials u(n,x): 1, 1 + 3x, 1 + 5x + 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

Cf. A209998, A208510.

Sequence in context: A094353 A298662 A306780 * A340804 A129801 A240222

Adjacent sequences: A209993 A209994 A209995 * A209997 A209998 A209999

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 23 2012

Status

approved