A207628 Triangle of coefficients of polynomials v(n,x) jointly generated with A207627; see the Formula section.

1, 1, 4, 1, 6, 8, 1, 8, 20, 16, 1, 10, 36, 56, 32, 1, 12, 56, 128, 144, 64, 1, 14, 80, 240, 400, 352, 128, 1, 16, 108, 400, 880, 1152, 832, 256, 1, 18, 140, 616, 1680, 2912, 3136, 1920, 512, 1, 20, 176, 896, 2912, 6272, 8960, 8192, 4352, 1024, 1, 22, 216

Offset: 1

Author

Clark Kimberling, Feb 21 2012

Keywords

• nonn
• tabl

Comments

Column n is divisible by 2^(n-1); row n ends with 2^(n-1) for n > 2.

Also triangle T(n,k), k=0..n, read by rows, given by (1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (4, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 25 2012

Also seems to be square array of unsigned coefficients of 3U-2T (with T and U the two sequences of Chebyshev polynomials). - Thomas Baruchel, Jun 03 2018

Examples

			First five rows:
  1;
  1,  4;
  1,  6,  8;
  1,  8, 20, 16;
  1, 10, 36, 56, 32;
		

Crossrefs

Cf. A207627.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x]
v[n_, x_] := 2 x*u[n - 1, x] + 2 x*v[n - 1, x] + 1
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A207627 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A207628 *)

Formula

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

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

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

The polynomials v(n,x) seem to be v(n)=sum(k=0,n-1, (-1)^(k+n+1) * x^k * polcoeff( 3*polchebyshev(2*n-k-2,2)-2*polchebyshev(2*n-k-2,1), k)) by using the PARI syntax. - Thomas Baruchel, Jun 05 2018

As triangle T(n,k), k=0..n:

G.f.: (1+2*y*x)/(1-(1+2*y)*x). - Philippe Deléham, Feb 25 2012

T(n,k) = 2*T(n-1,k-1) + T(n-1,k) with T(0,0) = T(1,0) = 1, T(1,1) = 4. - Philippe Deléham, Feb 25 2012

As triangle T(n,k), k=0..n, it is given by T(n,k) = A029635(n,k)*2^k with T(0,0) = 1. - Philippe Deléham, Feb 25 2012