A101343 Triangle read by rows: nonzero coefficients of the polynomials F_n(x) which express derivatives of tan(z) in terms of powers of tan(z).

1, 1, 1, 2, 2, 6, 8, 2, 24, 40, 16, 120, 240, 136, 16, 720, 1680, 1232, 272, 5040, 13440, 12096, 3968, 272, 40320, 120960, 129024, 56320, 7936, 362880, 1209600, 1491840, 814080, 176896, 7936, 3628800, 13305600, 18627840, 12207360, 3610112, 353792

Offset

0,4

Comments

Interpolates between factorials and tangent numbers.

References

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1998, p. 287.

Links

Alois P. Heinz, Rows n = 0..200, flattened

Dominique Foata and Guo-Niu Han, Multivariable Tangent and Secant q-derivative Polynomials, 2012. From N. J. A. Sloane, Oct 05 2012

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 [math.CO], 2021.

Formula

t(n,0)=n!; t(n,k)=tr(n,k)+tr(n,k-1), k<=n/2; t(n,floor((n+1)/2)-1)=tr(n,floor((n+1)/2)-1); tr(n,i)=((sum(j=0..2*i, binomial(j+n-2*i-1,n-2*i-1)*(j+n-2*i)!*2^(2*i-j)*(-1)^(j-i)*Stirling2(n,j+n-2*i)))). - Vladimir Kruchinin, May 27 2011

From Tom Copeland, Sep 30 2015: (Start)

Reversed rows signed and aerated are generated by [(1-x^2)D]^n x with D = d/dx, so exp(t(1-x^2)D) x = tanh(t + atanh(x)) is the e.g.f. of this reversed array (see A145271).

Reversed rows unsigned and aerated are generated by [(1+x^2)D]^n x, so exp(t(1+x^2)D) x = tan(t + atan(x)) = x + (1 +x^2)*t + (2x + 2x^3)*t^2/2! + (2 + 8x^2 + 6x^4)*t^3/3! + (16x + 40x^3 + 24x^5)*t^4/4! + ... is the e.g.f. for the matrix on p. 666 of the Knuth and Buckholtz link.

E.g.f. for this entry's aerated array 1 + (1 + x^2)*t + (2 + 2x^2)*t^2/2! + (6 + 8x^2 + 2x^4)*t^3/3! + (24 + 40^x^2 + 16x^4)*t^4/4! + ... = x * tan(t*x + atan(1/x)). (End)

From Fabián Pereyra, Apr 22 2022: (Start)

T(n,k) = (n-2k)*T(n-1,k) + (n-2k+2)*T(n-1,k-1).

E.g.f.: A(x,t) = sqrt(t)*(sqrt(t)*sin(x*sqrt(t))+cos(x*sqrt(t)))/ (sqrt(t)*cos(x*sqrt(t))-sin(x*sqrt(t))). (End)

Example

For example, D tan(z) = (tan(z))^2 + 1.
Array begins:
    1;
    1,   1;
    2,   2,
    6,   8,   2;
   24,  40,  16,
  120, 240, 136,  16;

Mathematica

row[n_] := CoefficientList[ Derivative[n][Tan][z] /. Tan -> t /. Sec -> (Sqrt[1+t[#]^2]&), t[z]] // DeleteCases[#, 0]& // Reverse; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 26 2013 *)

Prog

(Maxima)
T(n, k):=if k=0 then Tr(n, k) else if 2*k-1=n then Tr(n, k-1) else Tr(n, k)+Tr(n, k-1);
Tr(n, i):=((sum(binomial(j+n-2*i-1, n-2*i-1)*(j+n-2*i)!*2^(2*i-j)*(-1)^(j-i)*stirling2(n, j+n-2*i), j, 0, 2*i))); /* Vladimir Kruchinin, May 27 2011 */

Crossrefs

Reflection of triangle A008293.

Column k=0 gives A000142.

Row sums give A000831.

T(2n-1,n) gives A000182 (for n>=1).

Sequence in context: A106168 A106166 A374330 * A284748 A134457 A326479

Adjacent sequences: A101340 A101341 A101342 * A101344 A101345 A101346

Keyword

nonn,easy,tabf

Author

Don Knuth, Jan 28 2005

Extensions

More terms from Vladeta Jovovic and Ralf Stephan, Jan 30 2005

Status

approved