A209764 Triangle of coefficients of polynomials v(n,x) jointly generated with A209763; see the Formula section.

1, 2, 2, 3, 4, 4, 4, 8, 14, 8, 5, 14, 32, 34, 16, 6, 22, 62, 96, 86, 32, 7, 32, 108, 218, 280, 202, 64, 8, 44, 174, 432, 718, 760, 470, 128, 9, 58, 264, 778, 1584, 2194, 1992, 1066, 256, 10, 74, 382, 1304, 3142, 5360, 6382, 5048, 2390, 512, 11, 92, 532

Offset

1,2

Comments

Row n begins with n and ends with 2^(n-1).

Row sums: 1,4,11,34,101,304,911,2734,... A060925.

Alternating row sums: 1,0,3,2,5,4,7,6,... A060925.

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

Links

Table of n, a(n) for n=1..58.

Formula

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

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

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

Example

First five rows:
1
2...2
3...4....4
4...8....14...8
5...14...32...34...16
First three polynomials v(n,x): 1, 2 + 2x , 3 + 4x + 4x^2.

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + 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[%]    (* A209763 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209764 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A081250 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A060925 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A033999 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A004442*)

Crossrefs

Cf. A209663, A208510.

Sequence in context: A359897 A209698 A141525 * A071475 A358104 A343228

Adjacent sequences: A209761 A209762 A209763 * A209765 A209766 A209767

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 14 2012

Status

approved