A208336 Triangle of coefficients of polynomials u(n,x) jointly generated with A208337; see the Formula section.

1, 1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 4, 9, 10, 5, 1, 5, 14, 22, 20, 8, 1, 6, 20, 40, 51, 38, 13, 1, 7, 27, 65, 105, 111, 71, 21, 1, 8, 35, 98, 190, 256, 233, 130, 34, 1, 9, 44, 140, 315, 511, 594, 474, 235, 55, 1, 10, 54, 192, 490, 924, 1295, 1324, 942, 420, 89, 1, 11

Offset

1,5

Comments

coef. of x^(n-1) in u(n,x): A000045(n), Fibonacci numbers

coef. of x^(n-1) in v(n,x): A000045(n+1)

row sums, u(n,1): A000129

row sums, v(n,1): A001333

alternating row sums, u(n,-1): 1,0,1,0,1,0,1,0,1,0,...

alternating row sums, v(n,-1): 1,-1,1,-1,1,-1,1,-1,...

Links

Table of n, a(n) for n=1..68.

Formula

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

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

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

T(n,k) = A038137(n-1,k). - Philippe Deléham, Apr 05 2012

Example

First five rows:
1
1...1
1...2...2
1...3...5...3
1...4...9...10...5
First five polynomials u(n,x):
1
1 + x
1 + 2x + 2x^2
1 + 3x + 5x^2 + 3x^3
1 + 4x + 9x^2 + 10x^3 + 5x^4

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := (x + 1)*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[%]    (* A208336 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208337 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* u row sums *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* v row sums *)
Table[u[n, x] /. x -> -1, {n, 1, z}](* u alt. row sums *)
Table[v[n, x] /. x -> -1, {n, 1, z}](* v alt. row sums *)

Crossrefs

Apart from offsets the same as A038137.

Cf. A208337.

Sequence in context: A239830 A140767 A060850 * A038137 A073133 A106179

Adjacent sequences: A208333 A208334 A208335 * A208337 A208338 A208339

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 26 2012

Status

approved