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

Author

Clark Kimberling, Feb 26 2012

Keywords

• nonn
• tabl

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,...

Examples

			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
		

Crossrefs

Apart from offsets the same as A038137.

Cf. A208337.

Programs

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 *)

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