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A373504
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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).
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0
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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
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OFFSET
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0,5
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COMMENTS
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Related to Chebyshev polynomials of the first kind; see A123588.
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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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):
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MATHEMATICA
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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 *)
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CROSSREFS
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Triangle without column 0 gives A334009.
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KEYWORD
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AUTHOR
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STATUS
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approved
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