A209567 Triangle of coefficients of polynomials u(n,x) jointly generated with A209568; see the Formula section.

1, 1, 1, 2, 3, 1, 3, 7, 6, 1, 4, 13, 18, 10, 1, 5, 21, 41, 39, 15, 1, 6, 31, 79, 108, 75, 21, 1, 7, 43, 136, 245, 250, 132, 28, 1, 8, 57, 216, 486, 661, 524, 217, 36, 1, 9, 73, 323, 875, 1497, 1601, 1015, 338, 45, 1, 10, 91, 461, 1464, 3031, 4109, 3556, 1844

Offset

1,4

Comments

For n>1, row n begins with n and ends with 1.

For n>1, penultimate number in row n is (n-1)st triangular number.

Alternating row sums: 1,0,0,1,0,0,1,0,0,1,0,0,...

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)+v(n-1,x),

v(n,x)=x*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
1...1
2...3....1
3...7....6....1
4...13...18...10...1
First three polynomials v(n,x): 1, 1 + x, 2 + 3x + x^2.

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
v[n_, x_] := x*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[%]   (* A209567 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A209568 *)

Crossrefs

Cf. A209568, A208510.

Sequence in context: A236376 A063967 A059397 * A208338 A236918 A152821

Adjacent sequences: A209564 A209565 A209566 * A209568 A209569 A209570

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 10 2012

Status

approved