A209703 Triangle of coefficients of polynomials u(n,x) jointly generated with A209704; see the Formula section.

1, 0, 2, 0, 3, 3, 0, 4, 6, 5, 0, 5, 10, 14, 8, 0, 6, 15, 28, 28, 13, 0, 7, 21, 48, 66, 55, 21, 0, 8, 28, 75, 129, 149, 104, 34, 0, 9, 36, 110, 225, 326, 319, 193, 55, 0, 10, 45, 154, 363, 626, 774, 661, 352, 89, 0, 11, 55, 208, 553, 1099, 1625, 1761, 1332, 634

Offset

1,3

Comments

For n>1, row n begins with 0, has second term n, and ends with F(n+1), where F=A000045 (Fibonacci numbers); for n>2, column 2 consists of triangular numbers.

Row sums: A098790.

Alternating row sums: 1,-2,0,-3,-1,-4,-3,-5,-3,-6,,...

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

Links

Table of n, a(n) for n=1..65.

Formula

u(n,x)=x*u(n-1,x)+x*v(n-1,x),

v(n,x)=(x+1)*u(n-1,x)+v(n-1,x)+1,

where u(1,x)=1, v(1,x)=1.

Example

First five rows:
1
0...2
0...3....3
0...4....6...5
0...5...10...14...8
First three polynomials v(n,x): 1, 2x, 3x + 3x^2.

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := (x + 1)*u[n - 1, 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[%]  (* A209703 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A209704 *)

Crossrefs

Cf. A209704, A208510.

Sequence in context: A265208 A265020 A325191 * A279779 A279587 A127952

Adjacent sequences: A209700 A209701 A209702 * A209704 A209705 A209706

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 12 2012

Status

approved