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

1,4

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.

Links

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

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A209562, A208510.

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

Sequence in context: A208805 A208514 A179901 * A283822 A210789 A105809

Adjacent sequences: A209558 A209559 A209560 * A209562 A209563 A209564

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 10 2012

Status

approved