A210552 Triangle of coefficients of polynomials u(n,x) jointly generated with A210553; see the Formula section.

1, 1, 2, 1, 3, 3, 1, 4, 5, 5, 1, 5, 7, 10, 8, 1, 6, 9, 16, 18, 13, 1, 7, 11, 23, 31, 33, 21, 1, 8, 13, 31, 47, 62, 59, 34, 1, 9, 15, 40, 66, 101, 119, 105, 55, 1, 10, 17, 50, 88, 151, 205, 227, 185, 89, 1, 11, 19, 61, 113, 213, 321, 414, 426, 324, 144, 1, 12, 21, 73

Offset

1,3

Comments

Let T(n,k) denote the term in row n, column k.

T(n,n): A000045 (Fibonacci numbers)

T(n,n-1): A010049 (second-order Fibonacci numbers)

T(n,1): 1,1,1,1,1,1,1,1,1,1,1,,...

T(n,2): 2,3,4,5,6,7,8,9,10,11,...

T(n,3): 3,5,7,9,11,13,15,17,19,...

T(n,4): A052905

Row sums: A000225

Alternating row sums: A094024 (signed)

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

Links

Table of n, a(n) for n=1..70.

Formula

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

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

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

Example

First five rows:
1
1...2
1...3...3
1...4...5...5
1...5...7...10...8
First three polynomials u(n,x): 1, 1 + 2x, 1 + 3x + 3x^2.

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*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[%]   (* A210552 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210553 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A094024 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A052551 *)

Crossrefs

Cf. A210553, A208510.

Sequence in context: A210489 A344821 A125175 * A193376 A185095 A177888

Adjacent sequences: A210549 A210550 A210551 * A210553 A210554 A210555

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 22 2012

Status

approved