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

1,4

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

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) = 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

Example

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.

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 *)

Crossrefs

Cf. A209420, A208510.

Sequence in context: A214978 A295380 A093768 * A119011 A340440 A300866

Adjacent sequences: A209416 A209417 A209418 * A209420 A209421 A209422

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 09 2012

Status

approved