login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A008293 Triangle of coefficients in expansion of D^n (tan x) in powers of tan x. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..990

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.

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.

FORMULA

a(0, k) = delta(1, k); a(n+1, k) = (k-1)a(n, k-1)+(k+1)a(n, k+1).

EXAMPLE

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)

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-Fran├žois Alcover, Apr 05 2013 *)

CROSSREFS

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

Sequence in context: A085484 A116585 A230935 * A185811 A011140 A171715

Adjacent sequences:  A008290 A008291 A008292 * A008294 A008295 A008296

KEYWORD

nonn,tabf,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 23 13:07 EDT 2018. Contains 316528 sequences. (Running on oeis4.)