A210799 Triangle of coefficients of polynomials u(n,x) jointly generated with A210800; see the Formula section.

1, 3, 1, 5, 4, 2, 11, 13, 9, 3, 17, 32, 32, 17, 5, 35, 77, 96, 72, 32, 8, 53, 164, 254, 243, 153, 59, 13, 107, 353, 641, 739, 579, 313, 107, 21, 161, 704, 1496, 2042, 1938, 1305, 623, 192, 34, 323, 1433, 3440, 5348, 5898, 4774, 2831, 1213, 341, 55, 485

Offset

1,2

Comments

Row n starts with A060647(n) and ends with F(n), where F=A000045 (Fibonacci numbers).

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

Links

Table of n, a(n) for n=1..56.

Formula

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

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

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

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

Example

First five rows:
1
3....1
5....4....2
11...13...9....3
17...32...32...17...5
First three polynomials u(n,x): 1, 3 + x, 5 + 4x + 2x^2.

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 1; c = 1; h = 2; p = -1; f = 0;
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[%]   (* A210799 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210800 *)

Crossrefs

Cf. A210800, A208510.

Sequence in context: A210560 A208922 A209770 * A068512 A011090 A260629

Adjacent sequences: A210796 A210797 A210798 * A210800 A210801 A210802

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 27 2012

Status

approved