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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). 3
 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 (list; graph; refs; listen; history; text; internal format)
 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 Dominique Foata and Guo-Niu Han, Multivariable Tangent and Secant q-derivative Polynomials. - 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, Yeong-Nan Yeh, Alternating Eulerian polynomials and left peak polynomials, arXiv:2104.09374, 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) 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. Sequence in context: A283824 A106168 A106166 * 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

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Last modified June 20 13:10 EDT 2021. Contains 345164 sequences. (Running on oeis4.)