A208765 Triangle of coefficients of polynomials u(n,x) jointly generated with A208766; see the Formula section.

1, 1, 2, 1, 4, 6, 1, 6, 18, 14, 1, 8, 36, 56, 38, 1, 10, 60, 140, 190, 94, 1, 12, 90, 280, 570, 564, 246, 1, 14, 126, 490, 1330, 1974, 1722, 622, 1, 16, 168, 784, 2660, 5264, 6888, 4976, 1606, 1, 18, 216, 1176, 4788, 11844, 20664, 22392, 14454, 4094, 1

Offset

1,3

Comments

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

Subtriangle of the triangle given by (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 18 2012

Links

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Formula

u(n,x) = u(n-1,x) + 2*x*v(n-1,x),

v(n,x) = 2*x*u(n-1,x) + (x+1)*v(n-1,x),

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

From Philippe Deléham, Mar 18 2012: (Start)

As DELTA-triangle with 0 <= k <= n:

G.f.: (1-x-y*x+2*y*x^2-4*y^2*x^2)/(1-2*x-y*x+x^2+y*x^2-4*y^2*x^2).

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1) + 4*T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 2 and T(n,k) = 0 if k<0 or if k>n.

T(n,k) = binomial(n-1,k)*A026597(k). (End)

Example

First five rows:
  1;
  1,  2;
  1,  4,  6;
  1,  6, 18, 14;
  1,  8, 36, 56, 38;
First five polynomials u(n,x):
  1
  1 + 2x
  1 + 4x + 6x^2
  1 + 6x + 18x^2 + 14x^3
  1 + 8x + 36x^2 + 56x^3 + 38x^4
(1, 0, 0, 1, 0, 0, ...) DELTA (0, 2, 1, -2, 0, 0, ...) begins:
  1;
  1,  0;
  1,  2,  0;
  1,  4,  6,   0;
  1,  6, 18,  14,   0;
  1,  8, 36,  56,  38,  0;
  1, 10, 60, 140, 190, 94, 0. - Philippe Deléham, Mar 18 2012

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
v[n_, x_] := 2 x*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[%]     (* A208765 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]     (* A208766 *)
Rest[CoefficientList[CoefficientList[Series[(1-x-y*x+2*y*x^2-4*y^2*x^2)/( 1-2*x-y*x+x^2+y*x^2-4*y^2*x^2), {x, 0, 20}, {y, 0, 20}], x], y]//Flatten] (* G. C. Greubel, Mar 28 2018 *)

Crossrefs

Cf. A208766, A208510.

Sequence in context: A059369 A369518 A199530 * A232335 A098473 A121757

Adjacent sequences: A208762 A208763 A208764 * A208766 A208767 A208768

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 02 2012

Status

approved