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

1, 3, 1, 4, 3, 2, 5, 6, 8, 3, 6, 10, 18, 14, 5, 7, 15, 33, 38, 27, 8, 8, 21, 54, 81, 83, 49, 13, 9, 28, 82, 150, 197, 170, 89, 21, 10, 36, 118, 253, 401, 448, 342, 159, 34, 11, 45, 163, 399, 736, 999, 987, 671, 282, 55, 12, 55, 218, 598, 1253, 1988, 2387, 2106

Offset

1,2

Comments

For n>1, row n starts with n+1, followed by the n-th

triangular number, and ends in F(n+1), where F=A000045

(Fibonacci numbers).

Column 3: A166830.

Row sums: A048654.

Alternating row sums: 1,2,3,4,5,6,7,8,9,...

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

Links

Table of n, a(n) for n=1..63.

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
3...1
4...3....2
5...6....8....3
6...10...18...14...5
First three polynomials v(n,x): 1, 3 + x , 4 + 3x + 2x^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. A209703, A208510.

Sequence in context: A104764 A152842 A307280 * A339106 A082909 A335906

Adjacent sequences: A209701 A209702 A209703 * A209705 A209706 A209707

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 12 2012

Status

approved