A373504 Triangular array: row n gives the coefficients T(n,k) of powers x^(2k) in the series expansion of ((b^n + b^(-n))/2)^2, where b = x + sqrt(x^2 + 1).

1, 1, 1, 1, 4, 4, 1, 9, 24, 16, 1, 16, 80, 128, 64, 1, 25, 200, 560, 640, 256, 1, 36, 420, 1792, 3456, 3072, 1024, 1, 49, 784, 4704, 13440, 19712, 14336, 4096, 1, 64, 1344, 10752, 42240, 90112, 106496, 65536, 16384, 1, 81, 2160, 22176, 114048, 329472, 559104, 552960, 294912, 65536

Offset: 0

Author

Clark Kimberling, Aug 03 2024

Keywords

• nonn
• tabl

Comments

Related to Chebyshev polynomials of the first kind; see A123588.

Examples

			First 8 rows:
  1
  1    1
  1    4     4
  1    9    24     16
  1   16    80    128     64
  1   25   200    560    640    256
  1   36   420   1792   3456   3072   1024
  1   49   784   4704  13440  19612  14336  4096
The 4th polynomial is 1 + 9 x^2 + 24 x^4 + 16 x^6.
		

Crossrefs

Cf. A000012 (col 0), A000290 (col 1), A002415 ((1/4)*col(2)), A112742 (col 2), A000302 (T(n,n)), A123588, A008310.

Row sums give A055997(n+1).

Triangle without column 0 gives A334009.

Programs

Maple

p:= proc(n) option remember; (b-> series(
      ((b^n+b^(-n))/2)^2, x, 2*n+1))(x+sqrt(x^2+1))
    end:
T:= (n, k)-> coeff(p(n), x, 2*k):
seq(seq(T(n,k), k=0..n), n=0..10);  # Alois P. Heinz, Aug 03 2024

Mathematica

t[n_] := ((x + Sqrt[x^2 + 1])^n + (x + Sqrt[x^2 + 1])^(-n))/2
u = Expand[Table[FullSimplify[Expand[t[n]]], {n, 0, 10}]^2]
v = Column[CoefficientList[u, x^2]] (* array *)
Flatten[v] (* sequence *)
T[n_, k_] := If[k==0, 1, 4^(k - 1)*(2*Binomial[n + k, 2*k] - Binomial[n + k -1, 2*k -1])]; Flatten[Table[T[n,k],{n,0,9},{k,0,n}]] (* Detlef Meya, Aug 11 2024 *)

Formula

T(n, k) = if (k=0) then 1, otherwise 4^(k - 1)*(2*binomial(n + k, 2*k) - binomial(n + k - 1, 2*k - 1)). - Detlef Meya, Aug 11 2024