A209419 Triangle of coefficients of polynomials u(n,x) jointly generated with A209420; see the Formula section.

1, 1, 1, 2, 3, 1, 3, 8, 6, 1, 5, 17, 21, 10, 1, 8, 35, 58, 45, 15, 1, 13, 68, 144, 154, 85, 21, 1, 21, 129, 330, 452, 350, 147, 28, 1, 34, 239, 719, 1198, 1195, 714, 238, 36, 1, 55, 436, 1506, 2959, 3611, 2799, 1344, 366, 45, 1, 89, 785, 3063, 6930, 10005, 9537, 5985, 2376, 540, 55, 1

Offset: 1

Author

Clark Kimberling, Mar 09 2012

Keywords

• nonn
• tabl

Comments

Column 1: Fibonacci numbers (A000045)

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

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

Triangle given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 26 2012

Examples

			First five rows:
  1;
  1,  1;
  2,  3,  1;
  3,  8,  6,  1;
  5, 17, 21, 10,  1;
First three polynomials v(n,x): 1, 1 + x, 2 + 3x + x^2.
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

Cf. A209420, A208510.

Programs

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 + 1)*u[n - 1, x] + (x + 1)*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[%]    (* A209419 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209420 *)
CoefficientList[CoefficientList[Series[(1*x - x^2*y)/(1 - x - x^2 - 2*y*x + y^2*x^2), {x,0,10}, {y,0,10}], x], y] // Flatten (* G. C. Greubel, Jan 03 2018 *)

Formula

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

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

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

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-2), T(1,0) = T(2,0) = T(2,1) = 1, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 26 2012

G.f.: x*(1 - x*y)/(1 - x - x^2 - 2*y*x + y^2*x^2). - G. C. Greubel, Jan 03 2018