A207636 Triangle of coefficients of polynomials v(n,x) jointly generated with A207635; see Formula section.

1, 3, 2, 6, 7, 2, 12, 20, 11, 2, 24, 52, 42, 15, 2, 48, 128, 136, 72, 19, 2, 96, 304, 400, 280, 110, 23, 2, 192, 704, 1104, 960, 500, 156, 27, 2, 384, 1600, 2912, 3024, 1960, 812, 210, 31, 2, 768, 3584, 7424, 8960, 6944, 3584, 1232, 272, 35, 2, 1536, 7936

Offset

1,2

Comments

As triangle T(n,k) with 0 <= k <= n, it is (3, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 26 2012

Links

Table of n, a(n) for n=1..57.

Formula

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

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, Feb 26 2012: (Start)

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

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

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

Sum_{k=0..n} T(n,k)*x^k = A003945(n), |A084244(n)|, A189274(n) for x = 0, 1, 3 respectively.

Sum_{k=0..n} T(n,k)*x^(n-k) = A040000(n), |A084244(n)|, A128625(n) for x = 0, 1, 2 respectively. (End)

Example

First five rows:
   1;
   3,  2;
   6,  7,  2;
  12, 20, 11,  2;
  24, 52, 42, 15,  2;

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x]
v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A207635 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A207636 *)

Crossrefs

Cf. A207635.

Cf. A084938, A003945, A040000.

Sequence in context: A102004 A233208 A196518 * A125764 A023897 A267100

Adjacent sequences: A207633 A207634 A207635 * A207637 A207638 A207639

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 24 2012

Status

approved