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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).
4

%I #66 Jul 08 2024 16:46:22

%S 1,1,1,2,2,6,8,2,24,40,16,120,240,136,16,720,1680,1232,272,5040,13440,

%T 12096,3968,272,40320,120960,129024,56320,7936,362880,1209600,1491840,

%U 814080,176896,7936,3628800,13305600,18627840,12207360,3610112,353792

%N 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).

%C Interpolates between factorials and tangent numbers.

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

%H Alois P. Heinz, <a href="/A101343/b101343.txt">Rows n = 0..200, flattened</a>

%H Dominique Foata and Guo-Niu Han, <a href="http://www-irma.u-strasbg.fr/~foata/paper/pub119derivative.pdf">Multivariable Tangent and Secant q-derivative Polynomials</a>, 2012. From _N. J. A. Sloane_, Oct 05 2012

%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, and Yeong-Nan Yeh, <a href="https://arxiv.org/abs/2104.09374">Alternating Eulerian polynomials and left peak polynomials</a>, arXiv:2104.09374 [math.CO], 2021.

%F 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

%F From _Tom Copeland_, Sep 30 2015: (Start)

%F 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).

%F 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.

%F 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)

%F From _Fabián Pereyra_, Apr 22 2022: (Start)

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

%F 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)

%e For example, D tan(z) = (tan(z))^2 + 1.

%e Array begins:

%e 1;

%e 1, 1;

%e 2, 2,

%e 6, 8, 2;

%e 24, 40, 16,

%e 120, 240, 136, 16;

%t 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 *)

%o (Maxima)

%o 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);

%o 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 */

%Y Reflection of triangle A008293.

%Y Column k=0 gives A000142.

%Y Row sums give A000831.

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

%K nonn,easy,tabf

%O 0,4

%A _Don Knuth_, Jan 28 2005

%E More terms from _Vladeta Jovovic_ and _Ralf Stephan_, Jan 30 2005