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

1, 2, 2, 3, 6, 3, 5, 14, 13, 4, 8, 30, 41, 24, 5, 13, 60, 109, 96, 40, 6, 21, 116, 262, 308, 196, 62, 7, 34, 218, 590, 868, 743, 364, 91, 8, 55, 402, 1267, 2240, 2413, 1604, 630, 128, 9, 89, 730, 2627, 5424, 7046, 5926, 3186, 1032, 174, 10, 144, 1310, 5299, 12516, 19040, 19382, 13255, 5928, 1617, 230

Offset

1,2

Comments

Column 1: Fibonacci numbers (A000045).

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

Triangle given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 26 2012

Links

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

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

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

Example

First five rows:
  1;
  2,  2;
  3,  6,  3;
  5, 14, 13,  4;
  8, 30, 41, 24,  5;
First three polynomials v(n,x): 1, 2 + 2x, 3 + 6x + 3x^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 + 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[x*(1 + x)/(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 *)

Crossrefs

Cf. A209419, A208510.

Sequence in context: A308503 A196967 A210859 * A317449 A222310 A294033

Adjacent sequences: A209417 A209418 A209419 * A209421 A209422 A209423

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 09 2012

Status

approved