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A101343
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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).
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4
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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
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OFFSET
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0,4
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COMMENTS
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Interpolates between factorials and tangent numbers.
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REFERENCES
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R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1998, p. 287.
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LINKS
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FORMULA
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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
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)
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)
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EXAMPLE
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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;
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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T(2n-1,n) gives A000182 (for n>=1).
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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