A136663 Triangle of coefficients of the Pascal sum of A053120 Chebyshev's T(n, x) polynomials :p(x,n)=2*x*p(x,n-1)-p(x,n-2); pp(x,n)=Sum[Binomial[n,m]*p(x,m),{m,0,n}].
1, 1, 1, 0, 2, 2, -2, 0, 6, 4, -4, -8, 4, 16, 8, -4, -20, -20, 20, 40, 16, 0, -24, -72, -40, 72, 96, 32, 8, 0, -112, -224, -56, 224, 224, 64, 16, 64, -32, -448, -624, 0, 640, 512, 128, 16, 144, 288, -288, -1584, -1584, 384, 1728, 1152, 256, 0, 160, 800, 960, -1600, -5088, -3680, 1920, 4480, 2560, 512
Offset: 1
Examples
{1},
{1, 1},
{0, 2, 2},
{-2, 0, 6, 4},
{-4, -8, 4, 16, 8},
{-4, -20, -20, 20, 40, 16},
{0, -24, -72, -40, 72, 96, 32},
{8, 0, -112, -224, -56, 224, 224, 64},
{16, 64, -32, -448, -624, 0, 640, 512, 128},
{16, 144, 288, -288, -1584, -1584, 384, 1728, 1152, 256},
{0,160, 800, 960, -1600, -5088, -3680, 1920, 4480, 2560, 512}
Crossrefs
Cf. A053120.
Programs
-
Mathematica
P[x, 0] = 1; P[x, 1] = x; P[x_, n_] := P[x, n] = 2*x*P[x, n - 1] - P[x, n - 2]; Q[x_, n_] := Q[x, n] = Sum[P[x, m]*Binomial[n, m], {m, 0, n}]; a = Table[CoefficientList[Q[x, n], x], {n, 0, 10}]; Flatten[a]
Formula
p(x,n)=2*x*p(x,n-1)-p(x,n-2); pp(x,n)=Sum[Binomial[n,m]*p(x,m),{m,0,n}]
Comments