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
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
Comments