A209132 Triangle of coefficients of polynomials v(n,x) jointly generated with A209131; see the Formula section.

1, 0, 3, 0, 4, 5, 0, 4, 12, 11, 0, 4, 20, 36, 21, 0, 4, 28, 76, 92, 43, 0, 4, 36, 132, 244, 228, 85, 0, 4, 44, 204, 508, 716, 540, 171, 0, 4, 52, 292, 916, 1732, 1972, 1252, 341, 0, 4, 60, 396, 1500, 3564, 5436, 5196, 2844, 683, 0, 4, 68, 516, 2292, 6564

Offset

1,3

Comments

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

As triangle T(n,k) with 0 <= k <= n, it is (0, 4/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -4/3, -2/3, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 21 2012

Row sums are powers of 3 (A000244). - Philippe Deléham, Mar 21 2012

Links

Table of n, a(n) for n=1..61.

Formula

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

v(n,x) = 2x*u(n-1,x) + x*v(n-1,x),

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

T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + 2*T(n-2,k-2), T(1,0) = 1, T(2,0) = 0, T(2,1) = 3 and T(n,k) = 0 if k < 0 or if k >= n.- Philippe Deléham, Mar 21 2012

G.f.: (-1-2*x*y+x)*x*y/((1+x*y)*(2*x*y+x-1)). - R. J. Mathar, Aug 12 2015

Example

First five rows:
  1;
  0,  3;
  0,  4,  5;
  0,  4, 12, 11;
  0,  4, 20, 36, 21;
First three polynomials v(n,x):
  1
     3x
     4x + 5x^2.

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + 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[%]    (* A209131 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209132 *)

Crossrefs

Cf. A209131, A208510.

Sequence in context: A240966 A308109 A354783 * A019789 A137335 A011077

Adjacent sequences: A209129 A209130 A209131 * A209133 A209134 A209135

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 05 2012

Status

approved