A209561 Triangle of coefficients of polynomials u(n,x) jointly generated with A209562; see the Formula section.

1, 1, 1, 2, 2, 1, 3, 4, 3, 1, 4, 7, 7, 4, 1, 5, 11, 14, 11, 5, 1, 6, 16, 25, 25, 16, 6, 1, 7, 22, 41, 50, 41, 22, 7, 1, 8, 29, 63, 91, 91, 63, 29, 8, 1, 9, 37, 92, 154, 182, 154, 92, 37, 9, 1, 10, 46, 129, 246, 336, 336, 246, 129, 46, 10, 1, 11, 56, 175, 375, 582, 672

Offset: 1

Author

Clark Kimberling, Mar 10 2012

Keywords

• nonn
• tabl

Comments

Alternating row sums: 1,0,1,1,1,1,1,1,1,1,1,1,1,1,...

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

Examples

			First five rows:
1
1...1
2...2...1
3...4...3...1
4...7...7...4...1
First three polynomials v(n,x): 1, 1 + x, 2 + 2x + x^2.
		

Links

• Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Crossrefs

Cf. A209562, A208510.

Cf. A083329 (row sums), A097613 (central terms).

Programs

Haskell

a209561 n k = a209561_tabl !! (n-1) !! (k-1)
a209561_row n = a209561_tabl !! (n-1)
a209561_tabl = [1] : iterate
               (\row -> zipWith (+) ([1] ++ row) (row ++ [0])) [1,1]
-- Reinhard Zumkeller, Dec 26 2012

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
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[%]   (* A209561 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A209562 *)

Formula

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

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

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

T(n,n) = 1; T(n,k) = A051597(n-2,k-1), 1 <= k < n. - Reinhard Zumkeller, Dec 26 2012