A208761 Triangle of coefficients of polynomials u(n,x) jointly generated with A208762; see the Formula section.

1, 1, 2, 1, 6, 4, 1, 12, 18, 8, 1, 20, 52, 50, 16, 1, 30, 120, 186, 126, 32, 1, 42, 240, 534, 576, 306, 64, 1, 56, 434, 1302, 1986, 1654, 718, 128, 1, 72, 728, 2828, 5712, 6632, 4484, 1650, 256, 1, 90, 1152, 5628, 14436, 21912, 20508, 11682, 3726, 512

Offset

1,3

Comments

Alternating row sums: 1,-1,-1,-1,-1,-1,-1,-1,...

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

Subtriangle of the triangle given by [1,0,1,0,0,0,0,...] DELTA [0,2,0,-1,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 04 2012

Links

Table of n, a(n) for n=1..55.

Formula

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

Recurrence: T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) + T(n-2,k-1) + 2*T(n-2,k-2). - Philippe Deléham, Mar 04 2012

G.f.: (-1-x*y+x)*x*y/(-1+x*y+2*x+2*x^2*y^2+x^2*y-x^2). - R. J. Mathar, Aug 12 2015

Example

First five rows:
  1;
  1,  2;
  1,  6,  4;
  1, 12, 18,  8;
  1, 20, 52, 50, 16;
First five polynomials u(n,x):
  1
  1 +  2x
  1 +  6x +  4x^2
  1 + 12x + 18x^2 +  8x^3
  1 + 20x + 52x^2 + 50x^3 + 16x^4
From Philippe Deléham, Mar 04 2012: (Start)
Triangle (1, 0, 1, 0, 0, 0, ...) DELTA (0, 2, 0, -1, 0, 0, ...) begins:
  1;
  1,   0;
  1,   2,   0;
  1,   6,   4,   0;
  1,  12,  18,   8,   0;
  1,  20,  52,  50,  16,   0;
  1,  30, 120, 186, 126,  32,  0; (End)

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_] := (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[%]    (* A208761 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208762 *)

Crossrefs

Cf. A208762, A208510.

Sequence in context: A185045 A208913 A208911 * A123519 A167024 A114687

Adjacent sequences: A208758 A208759 A208760 * A208762 A208763 A208764

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 03 2012

Status

approved