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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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2
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Interpolates between factorials and tangent numbers.
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REFERENCES
| R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1998, p. 287.
Knuth, D. E. and Buckholtz, Thomas J., Computation of tangent, Euler and Bernoulli numbers. Math. Comp. 21 1967 663-688.
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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)))); [From Vladimir Kruchinin, May 27 2011 ]
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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
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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))); [From Vladimir Kruchinin kru(AT)ie.tusur.ru, May 27 2011 ]
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CROSSREFS
| Reflection of triangle A008293.
Sequence in context: A099577 A106168 A106166 * A134457 A092522 A116542
Adjacent sequences: A101340 A101341 A101342 * A101344 A101345 A101346
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KEYWORD
| nonn,easy,tabf
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AUTHOR
| D. E. Knuth, Jan 28 2005
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs) and R. Stephan, Jan 30 2005
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