A208747 Triangle of coefficients of polynomials u(n,x) jointly generated with A208748; see the Formula section.

1, 1, 2, 1, 2, 8, 1, 2, 12, 24, 1, 2, 16, 40, 80, 1, 2, 20, 56, 160, 256, 1, 2, 24, 72, 256, 576, 832, 1, 2, 28, 88, 368, 992, 2112, 2688, 1, 2, 32, 104, 496, 1504, 3968, 7552, 8704, 1, 2, 36, 120, 640, 2112, 6464, 15232, 26880, 28160, 1, 2, 40, 136, 800

Offset: 1

Author

Clark Kimberling, Mar 02 2012

Keywords

• nonn
• tabl

Comments

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

Subtriangle of the triangle T(n,k) given by (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 14 2012

Examples

			First five rows:
1
1...2
1...2...8
1...2...12...24
1...2...16...40...80
First five polynomials u(n,x):
1
1 + 2x
1 + 2x + 8x^2
1 + 2x + 12x^2 + 24x^3
1 + 2x + 16x^2 + 40x^3 + 80x^4
(1, 0, -1, 1, 0, 0, ...) DELTA (0, 0, 0, -2, 0, 0, ...) begins :
1
1, 0
1, 2, 0
1, 2, 8, 0
1, 2, 12, 24, 0
1, 2, 16, 40, 80, 0
1, 2, 20, 56, 160, 256, 0
1, 2, 24, 72, 256, 576, 832, 0. - _Philippe Deléham_, Mar 14 2012
		

Crossrefs

Cf. A208748, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + 2 x*v[n - 1, x];
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[%]    (* A208747 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208748 *)

Formula

u(n,x)=u(n-1,x)+2x*v(n-1,x),

v(n,x)=2x*u(n-1,x)+2x*v(n-1,x),

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

T(n,k) = A208342(n,k)*2^k. - Philippe Deléham, Mar 05 2012

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - 2*T(n-2,k-1) + 4*T(n-2,k-2), T(1,0) = T(2,0) = 1, T(2,1) = 2, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 14 2012

G.f.: -x*y/(-1+2*x*y-2*x^2*y+4*x^2*y^2+x). - R. J. Mathar, Aug 11 2015