A208334 Triangle of coefficients of polynomials u(n,x) jointly generated with A208335; see the Formula section.

1, 1, 1, 1, 3, 1, 1, 6, 4, 1, 1, 10, 11, 6, 1, 1, 15, 25, 21, 7, 1, 1, 21, 50, 57, 30, 9, 1, 1, 28, 91, 133, 99, 45, 10, 1, 1, 36, 154, 280, 275, 168, 58, 12, 1, 1, 45, 246, 546, 675, 523, 250, 78, 13, 1, 1, 55, 375, 1002, 1509, 1433, 885, 370, 95, 15, 1, 1, 66, 550

Offset

1,5

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 (1, 0, 1, 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

Up to reflection at the vertical axis, the triangle of numbers given here coincides with the triangle given in A209415, i.e., the numbers are the same just read row-wise in the opposite direction. - Christine Bessenrodt, Jul 21 2012

Links

Table of n, a(n) for n=1..69.

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-y^2*x^2)/(1-2*x-y*x^2+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(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)

Example

First five rows:
  1;
  1,  1;
  1,  3,  1;
  1,  6,  4, 1;
  1, 10, 11, 6, 1;
First five polynomials u(n,x):
  1;
  1 +   x;
  1 +  3x +   x^2;
  1 +  6x +  4x^2 +  x^3;
  1 + 10x + 11x^2 + 6x^3 + x^4;
From Philippe Deléham, Mar 26 2012: (Start)
(1, 0, 1, 0, 0, 0, ...) DELTA (0, 1, 0, -1, 0, 0, ...) begins:
  1;
  1,  0;
  1,  1,  0;
  1,  3,  1,  0;
  1,  6,  4,  1,  0;
  1, 10, 11,  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. A208335, A209415.

Sequence in context: A271665 A184049 A125230 * A162430 A305059 A355996

Adjacent sequences: A208331 A208332 A208333 * A208335 A208336 A208337

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 26 2012

Status

approved