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
Examples
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;
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.
Crossrefs
Programs
-
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 *) -
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 */
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)
Extensions
More terms from Vladeta Jovovic and Ralf Stephan, Jan 30 2005
Comments