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A008293
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Triangle of coefficients in expansion of D^n (tan x) in powers of tan x.
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5
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1, 1, 1, 2, 2, 2, 8, 6, 16, 40, 24, 16, 136, 240, 120, 272, 1232, 1680, 720, 272, 3968, 12096, 13440, 5040, 7936, 56320, 129024, 120960, 40320, 7936, 176896, 814080, 1491840, 1209600, 362880, 353792, 3610112, 12207360, 18627840, 13305600, 3628800
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OFFSET
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0,4
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COMMENTS
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James Gregory calculated the first eight rows of this table (with some numerical errors) in 1671. See Roy, p. 299. - Peter Bala, Sep 06 2016
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LINKS
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FORMULA
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T(0, k) = delta(1, k), T(n, k) = (k-1)*T(n-1, k-1) + (k+1)*T(n-1, k+1).
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EXAMPLE
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Table begins
1
1 1
2 2
2 8 6
16 40 24
16 136 240 120
272 1232 1680 720
272 3968 12096 13440 5040
...
D(tan(x)) = 1 + tan(x)^2.
D^2(tan(x)) = 2*tan(x) + 2*tan(x)^3.
D^3(tan(x)) = 2 + 8*tan(x)^2 + 6*tan(x)^4.
D^4(tan(x)) = 16*tan(x) + 40*tan(x)^3 + 24*tan(x)^5. (End)
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MATHEMATICA
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row[n_] := CoefficientList[ D[Tan[x], {x, n}] /. Tan -> Identity /. Sec -> Function[Sqrt[1 + #^2]], x] // DeleteCases[#, 0]&; Table[row[n], {n, 0, 10}] // Flatten // Prepend[#, 1] & (* Jean-François Alcover, Apr 05 2013 *)
T[ n_, k_] := If[n<1, Boole[n==0 && k==1], (k-1)*T[n-1, k-1] + (k+1)*T[n-1, k+1]]; (* Michael Somos, Jul 08 2024 *)
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PROG
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(PARI) {T(n, k) = if(n<1, n==0 && k==1, (k-1)*T(n-1, k-1) + (k+1)*T(n-1, k+1))}; /* Michael Somos, Jul 08 2024 */
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CROSSREFS
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Bisection of column k=0 gives A000182.
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KEYWORD
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nonn,tabf,easy,nice
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AUTHOR
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STATUS
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approved
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