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

1,2

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

Links

Table of n, a(n) for n=1..55.

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)

Example

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)

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 *)

Crossrefs

Cf. A013609, A208510, A110813.

Sequence in context: A367208 A249741 A246275 * A208932 A189951 A209776

Adjacent sequences: A209754 A209755 A209756 * A209758 A209759 A209760

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 23 2012

Status

approved