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

Author

Clark Kimberling, Feb 26 2012

Keywords

• nonn
• tabl

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

Examples

			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)
		

Crossrefs

Cf. A208335, A209415.

Programs

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 *)

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)