A209757 Triangle of coefficients of polynomials v(n,x) jointly generated with A013609; see the Formula section.

1, 3, 2, 5, 8, 4, 7, 18, 20, 8, 9, 32, 56, 48, 16, 11, 50, 120, 160, 112, 32, 13, 72, 220, 400, 432, 256, 64, 15, 98, 364, 840, 1232, 1120, 576, 128, 17, 128, 560, 1568, 2912, 3584, 2816, 1280, 256, 19, 162, 816, 2688, 6048, 9408, 9984, 6912, 2816, 512

Offset: 1

Author

Clark Kimberling, Mar 23 2012

Keywords

• nonn
• tabl

Comments

For a discussion and guide to related arrays, see A208510.

Subtriangle of the triangle given by (1, 2, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 24 2012

Examples

			First five rows:
  1;
  3,  2;
  5,  8,  4;
  7, 18, 20,  8;
  9, 32, 56, 48, 16;
First three polynomials v(n,x):
  1
  3 + 2x
  5 + 8x + 4x^2.
From _Philippe Deléham_, Mar 24 2012: (Start)
(1, 2, -2, 1, 0, 0, ...) DELTA (0, 2, 0, 0, 0, ...) begins:
  1;
  1,  0;
  3,  2,  0;
  5,  8,  4,  0;
  7, 18, 20,  8,  0;
  9, 32, 56, 48, 16,  0; (End)
		

Crossrefs

Cf. A013609, A208510, A110813.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
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[%]    (* A013609 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209757 *)

Formula

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

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

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

From Philippe Deléham, Mar 24 2012: (Start)

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

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

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

T(n,k) = 2^k*binomial(n-1,k)*(2*n-k-1)/(k+1). (End)

From Peter Bala, Dec 21 2014: (Start)

Following remarks assume an offset of 0.

T(n,k) = 2^k * A110813(n,k).

Riordan array ((1+x)/(1-x)^2, 2*x/(1-x)).

exp(2*x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(2*x)*(7 + 18*x + 20*x^2/2! + 8*x^3/3!) = 7 + 32*x + 120*x^2/2! + 400*x^3/3! + 1232*x^4/4! + .... The same property holds more generally for Riordan arrays of the form (f(x), 2*x/(1-x)). (End)