A208339 Triangle of coefficients of polynomials v(n,x) jointly generated with A208838; see the Formula section.

1, 1, 3, 1, 4, 7, 1, 5, 13, 17, 1, 6, 20, 40, 41, 1, 7, 28, 72, 117, 99, 1, 8, 37, 114, 241, 332, 239, 1, 9, 47, 167, 425, 769, 921, 577, 1, 10, 58, 232, 682, 1492, 2368, 2512, 1393, 1, 11, 70, 310, 1026, 2598, 5008, 7096, 6761, 3363, 1, 12, 83, 402, 1472

Offset

1,3

Comments

Subtriangle of the triangle given by (1, 0, -2/3, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -2/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 27 2012

Links

Table of n, a(n) for n=1..60.

Formula

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

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

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

Contribution from Philippe Deléham, Mar 27 2012: (Start)

As DELTA-triangle T(n,k) with 0<=k<=n:

G.f.: (1-2*y*x+2*y*x^2-y^2*x^2)/(1-x-2*y*x+y*x^2-y^2*x^2).

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(2,1) = 3, T(1,1) = T(2,2) = 0 nd T(n,k) = 0 if k<0 or if k>n. (End)

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

Example

First five rows:
1
1...3
1...4...7
1...5...13...17
1...6...20...40...41
First five polynomials v(n,x):
1
1 + 3x
1 + 4x + 7x^2
1 + 5x + 13x^2 + 17x^3
1 + 6x + 20x^2 + 40x^3 + 41x^4
Contribution from Philippe Deléham, Mar 27 2012: (Start)
(1, 0, -2/3, 2/3, 0, 0,...) DELTA (0, 3, -2/3, -1/3, 0, 0,...) begins :
1
1, 0
1, 3, 0
1, 4, 7, 0
1, 5, 13, 17, 0
1, 6, 20, 40, 41, 0. (End)

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := (x + 1)*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[%]   (* A208338 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A208339 *)

Crossrefs

Cf. A208338.

Sequence in context: A054143 A104746 A350584 * A328463 A185722 A287376

Adjacent sequences: A208336 A208337 A208338 * A208340 A208341 A208342

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 27 2012

Status

approved