A209148 Triangle of coefficients of polynomials u(n,x) jointly generated with A209149; see the Formula section.

1, 2, 1, 5, 5, 1, 11, 16, 7, 1, 23, 44, 30, 9, 1, 47, 112, 104, 48, 11, 1, 95, 272, 320, 200, 70, 13, 1, 191, 640, 912, 720, 340, 96, 15, 1, 383, 1472, 2464, 2352, 1400, 532, 126, 17, 1, 767, 3328, 6400, 7168, 5152, 2464, 784, 160, 19, 1, 1535, 7424

Offset: 1

Author

Clark Kimberling, Mar 07 2012

Keywords

• nonn
• tabl

Comments

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

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

Examples

			First five rows:
  1
  2....1
  5....5....1
  11...16...7....1
  23...44...30...9...1
First three polynomials v(n,x): 1, 2 + x, 5 + 5x + x^2.
		

Crossrefs

Cf. A209149, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, 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[%]    (* A209148 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209149 *)

Formula

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

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

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

T(n,k) = 3*T(n-1,k)+T(n-1,k-1)-2*T(n-2,k)-T(n-2,k-1), T(1,0)=T(2,1)=T(3,2)=1, T(2,0)=2, T(3,0)=T(3,1)=5, T(n,k)=0 if k<0 or if k>=n. - Philippe Deléham, Jan 17 2014

Extensions

Example corrected by Philippe Deléham, Jan 17 2014