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

1,2

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

Links

Table of n, a(n) for n=1..54.

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

Example

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

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 *)

Crossrefs

Cf. A208761, A208510.

Sequence in context: A222779 A222894 A210753 * A209746 A354950 A267822

Adjacent sequences: A208759 A208760 A208761 * A208763 A208764 A208765

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 03 2012

Status

approved