A008293 Triangle of coefficients in expansion of D^n (tan x) in powers of tan x.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabf
• easy
• nice

Comments

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

Examples

			From _Peter Bala_, Sep 06 2016: (Start)
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)
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..990
• William Y. C. Chen and Amy M. Fu, The Dumont Ansatz for the Eulerian Polynomials, Peak Polynomials and Derivative Polynomials, arXiv:2204.01497 [math.CO], 2022.
• M.-P. Grosset and A. P. Veselov, Bernoulli numbers and solitons, arXiv:math/0503175 [math.GM], 2005.
• Gordon Haigh, A "natural" approach to Pick's theorem, Math. Gaz. 64 (1980), no. 429, 173-180.
• Donald E. Knuth and Thomas J. Buckholtz, Computation of tangent, Euler and Bernoulli numbers, Math. Comp. 21 1967 663-688.
• Shi-Mei Ma, Qi Fang, Toufik Mansour, and Yeong-Nan Yeh, Alternating Eulerian polynomials and left peak polynomials, arXiv:2104.09374, 2021
• R. Roy, The Discovery of the Series Formula for Pi by Leibniz, Gregory and Nilakantha, Mathematics Magazine Vol. 63, No. 5 (Dec., 1990), 291-306.
• M. S. Tokmachev, Correlations Between Elements and Sequences in a Numerical Prism, Bulletin of the South Ural State University, Ser. Mathematics. Mechanics. Physics, 2019, Vol. 11, No. 1, 24-33.

Crossrefs

Cf. A008294. Other versions of same triangle: A101343, A155100.

T(n,ceiling(n/2)) gives A000142.

Bisection of column k=0 gives A000182.

Row sums give A000831.

Programs

Mathematica

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 *)

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 */

Formula

T(0, k) = delta(1, k), T(n, k) = (k-1)*T(n-1, k-1) + (k+1)*T(n-1, k+1).