A208335 Triangle of coefficients of polynomials v(n,x) jointly generated with A208834; see the Formula section.

1, 2, 1, 3, 3, 1, 4, 7, 5, 1, 5, 14, 15, 6, 1, 6, 25, 36, 23, 8, 1, 7, 41, 76, 69, 36, 9, 1, 8, 63, 147, 176, 123, 48, 11, 1, 9, 92, 266, 400, 355, 192, 66, 12, 1, 10, 129, 456, 834, 910, 635, 292, 82, 14, 1, 11, 175, 747, 1626, 2131, 1833, 1065, 410, 105, 15, 1

Offset

1,2

Comments

row sums, u(n,1): A000129

row sums, v(n,1): A001333

alternating row sums, u(n,-1): 1,0,-1,-2,-3,-4,-5,-6,...

alternating row sums, v(n,-1): 1,1,1,1,1,1,1,1,1,1,1,...

Subtriangle of the triangle T(n,k) given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 26 2012

Links

Table of n, a(n) for n=1..66.

Formula

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

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

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

From Philippe Deléham, Mar 26 2012: (Start)

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

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

T(n,k) = 2*T(n-1,k) - T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(2,0) = 2, T(1,1) = T(2,2) = 0. (End)

Example

First five rows:
  1;
  2,  1;
  3,  3,  1;
  4,  7,  5,  1;
  5, 14, 15,  6,  1;
First five polynomials v(n,x):
  1
  2 +   x
  3 +  3x +   x^2
  4 +  7x +  5x^2 +  x^3
  5 + 14x + 15x^2 + 6x^3 + x^4
From Philippe Deléham, Mar 26 2012: (Start)
(1, 1, -1, 1, 0, 0, 0, ...) DELTA (0, 1, 0, -1, 0, 0, ...) begins:
  1;
  1,  0;
  2,  1,  0;
  3,  3,  1,  0;
  4,  7,  5,  1,  0;
  5, 14, 15,  6,  1,  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] + 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[%]  (* A208334 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208335  *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* u row sums *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* v row sums *)
Table[u[n, x] /. x -> -1, {n, 1, z}](* u alt. row sums *)
Table[v[n, x] /. x -> -1, {n, 1, z}](* v alt. row sums *)

Crossrefs

Cf. A208334.

Sequence in context: A185943 A352001 A208337 * A208597 A179943 A089944

Adjacent sequences: A208332 A208333 A208334 * A208336 A208337 A208338

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 26 2012

Status

approved