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%I #45 Apr 22 2022 19:05:54
%S 1,1,1,2,2,2,8,6,16,40,24,16,136,240,120,272,1232,1680,720,272,3968,
%T 12096,13440,5040,7936,56320,129024,120960,40320,7936,176896,814080,
%U 1491840,1209600,362880,353792,3610112,12207360,18627840,13305600
%N Triangle of coefficients in expansion of D^n (tan x) in powers of tan x.
%C 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
%H Vincenzo Librandi, <a href="/A008293/b008293.txt">Table of n, a(n) for n = 0..990</a>
%H William Y. C. Chen and Amy M. Fu, <a href="https://arxiv.org/abs/2204.01497">The Dumont Ansatz for the Eulerian Polynomials, Peak Polynomials and Derivative Polynomials</a>, arXiv:2204.01497 [math.CO], 2022.
%H M.-P. Grosset and A. P. Veselov, <a href="https://arxiv.org/abs/math/0503175">Bernoulli numbers and solitons</a>, arXiv:math/0503175 [math.GM], 2005.
%H Gordon Haigh, <a href="http://www.jstor.org/stable/3615119">A "natural" approach to Pick's theorem</a>, Math. Gaz. 64 (1980), no. 429, 173-180.
%H Donald E. Knuth and Thomas J. Buckholtz, <a href="http://dx.doi.org/10.1090/S0025-5718-1967-0221735-9">Computation of tangent, Euler and Bernoulli numbers</a>, Math. Comp. 21 1967 663-688.
%H Shi-Mei Ma, Qi Fang, Toufik Mansour, Yeong-Nan Yeh, <a href="https://arxiv.org/abs/2104.09374">Alternating Eulerian polynomials and left peak polynomials</a>, arXiv:2104.09374, 2021
%H R. Roy, <a href="https://www.jstor.org/stable/2690896">The Discovery of the Series Formula for Pi by Leibniz, Gregory and Nilakantha</a>, Mathematics Magazine Vol. 63, No. 5 (Dec., 1990), 291-306.
%H M. S. Tokmachev, <a href="https://vestnik.susu.ru/mmph/article/viewFile/8337/6806">Correlations Between Elements and Sequences in a Numerical Prism</a>, Bulletin of the South Ural State University, Ser. Mathematics. Mechanics. Physics, 2019, Vol. 11, No. 1, 24-33.
%F T(0, k) = delta(1, k), T(n, k) = (k-1)*T(n-1, k-1) + (k+1)*T(n-1, k+1).
%e From _Peter Bala_, Sep 06 2016: (Start)
%e Table begins
%e 1
%e 1 1
%e 2 2
%e 2 8 6
%e 16 40 24
%e 16 136 240 120
%e 272 1232 1680 720
%e 272 3968 12096 13440 5040
%e ...
%e D(tan(x)) = 1 + tan(x)^2.
%e D^2(tan(x)) = 2*tan(x) + 2*tan(x)^3.
%e D^3(tan(x)) = 2 + 8*tan(x)^2 + 6*tan(x)^4.
%e D^4(tan(x)) = 16*tan(x) + 40*tan(x)^3 + 24*tan(x)^5. (End)
%t 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 *)
%Y Cf. A008294. Other versions of same triangle: A101343, A155100.
%K nonn,tabf,easy,nice
%O 0,4
%A _N. J. A. Sloane_
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