A208762 Triangle of coefficients of polynomials v(n,x) jointly generated with A208761; see the Formula section.

1, 2, 2, 3, 7, 4, 4, 17, 21, 8, 5, 34, 68, 55, 16, 6, 60, 174, 225, 137, 32, 7, 97, 384, 705, 674, 327, 64, 8, 147, 763, 1863, 2489, 1883, 761, 128, 9, 212, 1400, 4362, 7640, 8012, 5016, 1735, 256, 10, 294, 2412, 9318, 20542, 27996, 24144, 12885, 3897

Offset: 1

Author

Clark Kimberling, Mar 03 2012

Keywords

• nonn
• tabl

Comments

Alternating row sums: 1,0,0,0,0,0,0,0,0,... For a discussion and guide to related arrays, see A208510.

As triangle T(n,k) with 0<=k<=n, it is (2, -1/2, 1/2, 0 0 0 0 0 0 0 ...) DELTA (2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 04 2012

Row sums are in A055099. - Philippe Deléham, Mar 04 2012

Examples

			First five rows:
1
2...2
3...7....4
4...17...21...8
5...34...68...55...16
First five polynomials v(n,x):
1
2 + 2x
3 + 7x + 4x^2
4 + 17x + 22x^2 + 8x^3
5 + 34x + 68x^2 + 55x^3 + 16x^4
		

Crossrefs

Cf. A208761, 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_] := (x + 1)*u[n - 1, x] + (x + 1) 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[%]    (* A208761 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208762 *)

Formula

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

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

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

As triangle T(n,k) with 0<=k<=n : T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) + T(n-2,k-1) + 2*T(n-2,k-2), T(0,0) = 1, T(1,0) = T(1,1) = 2 and T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, Mar 04 2012

G.f.: (-1-x*y)*x*y/(-1+x*y+x^2*y+2*x^2*y^2+2*x-x^2). - R. J. Mathar, Aug 12 2015