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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).

%I A101343 #66 Jul 08 2024 16:46:22
%S A101343 1,1,1,2,2,6,8,2,24,40,16,120,240,136,16,720,1680,1232,272,5040,13440,
%T A101343 12096,3968,272,40320,120960,129024,56320,7936,362880,1209600,1491840,
%U A101343 814080,176896,7936,3628800,13305600,18627840,12207360,3610112,353792
%N 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).
%C A101343 Interpolates between factorials and tangent numbers.
%D A101343 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1998, p. 287.
%H A101343 Alois P. Heinz, <a href="/A101343/b101343.txt">Rows n = 0..200, flattened</a>
%H A101343 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 A101343 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 A101343 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 A101343 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 A101343 From _Tom Copeland_, Sep 30 2015: (Start)
%F A101343 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 A101343 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 A101343 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 A101343 From _Fabián Pereyra_, Apr 22 2022: (Start)
%F A101343 T(n,k) = (n-2k)*T(n-1,k) + (n-2k+2)*T(n-1,k-1).
%F A101343 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 A101343 For example, D tan(z) = (tan(z))^2 + 1.
%e A101343 Array begins:
%e A101343     1;
%e A101343     1,   1;
%e A101343     2,   2,
%e A101343     6,   8,   2;
%e A101343    24,  40,  16,
%e A101343   120, 240, 136,  16;
%t A101343 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 A101343 (Maxima)
%o A101343 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 A101343 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 A101343 Reflection of triangle A008293.
%Y A101343 Column k=0 gives A000142.
%Y A101343 Row sums give A000831.
%Y A101343 T(2n-1,n) gives A000182 (for n>=1).
%K A101343 nonn,easy,tabf
%O A101343 0,4
%A A101343 _Don Knuth_, Jan 28 2005
%E A101343 More terms from _Vladeta Jovovic_ and _Ralf Stephan_, Jan 30 2005